Solve for x cos(x)^4-sin(x)^4=cos(2x)

$\cos^{4} (x) - \sin^{4} (x) = \cos (2 x)$

Use the identity to solve the equation. In this identity, $θ$ represents the angle created by plotting point $(a, b)$ on a graph and therefore can be found using $θ = \tan^{-1} (\frac{b}{a})$ .

$a \sin (x) + b \cos (x) = R \sin (x + θ)$ where $R = \sqrt{a^{2} + b^{2}}$ and $θ = \tan^{-1} (\frac{b}{a})$

Set up the equation to find the value of $θ$ .

$\tan^{-1} (\frac{1}{- 1})$

Take the inverse tangent to solve the equation for $θ$ .

Tap for more steps…

Divide $1$ by $- 1$ .

$θ = \tan^{-1} (- 1)$

The exact value of $\tan^{-1} (- 1)$ is $- \frac{π}{4}$ .

$θ = - \frac{π}{4}$

Solve to find the value of $R$ .

Tap for more steps…

Raise $- 1$ to the power of $2$ .

$R = \sqrt{1 + {(1)}^{2}}$

One to any power is one.

$R = \sqrt{1 + 1}$

Add $1$ and $1$ .

$R = \sqrt{2}$

Substitute the known values into the equation.

$(\sqrt{2}) \sin (x - \frac{π}{4}) = \cos (2 x)$

Multiply $\sqrt{2}$ by $\sin (x - \frac{π}{4})$ .

$\sqrt{2} \sin (x - \frac{π}{4}) = \cos (2 x)$

Divide each term by $\sqrt{2}$ and simplify.

Tap for more steps…

Divide each term in $\sqrt{2} \sin (x - \frac{π}{4}) = \cos (2 x)$ by $\sqrt{2}$ .

$\frac{\sqrt{2} \sin (x - \frac{π}{4})}{\sqrt{2}} = \frac{\cos (2 x)}{\sqrt{2}}$

Cancel the common factor of $\sqrt{2}$ .

Tap for more steps…

Cancel the common factor.

$\frac{\sqrt{2} \sin (x - \frac{π}{4})}{\sqrt{2}} = \frac{\cos (2 x)}{\sqrt{2}}$

Divide $\sin (x - \frac{π}{4})$ by $1$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x)}{\sqrt{2}}$

Simplify $\frac{\cos (2 x)}{\sqrt{2}}$ .

Tap for more steps…

Multiply $\frac{\cos (2 x)}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x)}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Tap for more steps…

Multiply $\frac{\cos (2 x)}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps…

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{2}$

Evaluate the exponent.

$\sin (x - \frac{π}{4}) = \frac{\cos (2 x) \sqrt{2}}{2}$

Move all terms containing $x$ to the left side of the equation.

Tap for more steps…

Subtract $\frac{\cos (2 x) \sqrt{2}}{2}$ from both sides of the equation.

$\sin (x - \frac{π}{4}) - \frac{\cos (2 x) \sqrt{2}}{2} = 0$

Simplify the left side of the equation.

Tap for more steps…

Simplify the numerator.

Tap for more steps…

Use the double–angle identity to transform $\cos (2 x)$ to $\cos^{2} (x) - \sin^{2} (x)$ .

$\sin (x - \frac{π}{4}) - \frac{(\cos^{2} (x) - \sin^{2} (x)) \sqrt{2}}{2} = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = \sin (x)$ .

$\sin (x - \frac{π}{4}) - \frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x)) \sqrt{2}}{2} = 0$

To write $\frac{\sin (x - \frac{π}{4})}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sin (x - \frac{π}{4})}{1} \cdot \frac{2}{2} - \frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x)) \sqrt{2}}{2} = 0$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$\frac{\sin (x - \frac{π}{4}) \cdot 2}{1 \cdot 2} - \frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x)) \sqrt{2}}{2} = 0$

Multiply $2$ by $1$ .

$\frac{\sin (x - \frac{π}{4}) \cdot 2}{2} - \frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x)) \sqrt{2}}{2} = 0$

Combine the numerators over the common denominator.

$\frac{\sin (x - \frac{π}{4}) \cdot 2 - (\cos (x) + \sin (x)) (\cos (x) - \sin (x)) \sqrt{2}}{2} = 0$

Simplify the numerator.

Tap for more steps…

Move $2$ to the left of $\sin (x - \frac{π}{4})$ .

$\frac{2 \cdot \sin (x - \frac{π}{4}) - (\cos (x) + \sin (x)) (\cos (x) - \sin (x)) \sqrt{2}}{2} = 0$

Apply the distributive property.

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos (x) - \sin (x)) (\cos (x) - \sin (x)) \sqrt{2}}{2} = 0$

Expand $(- \cos (x) - \sin (x)) (\cos (x) - \sin (x))$ using the FOIL Method.

Tap for more steps…

Apply the distributive property.

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos (x) (\cos (x) - \sin (x)) - \sin (x) (\cos (x) - \sin (x))) \sqrt{2}}{2} = 0$

Apply the distributive property.

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos (x) \cos (x) - \cos (x) (- \sin (x)) - \sin (x) (\cos (x) - \sin (x))) \sqrt{2}}{2} = 0$

Apply the distributive property.

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos (x) \cos (x) - \cos (x) (- \sin (x)) - \sin (x) \cos (x) - \sin (x) (- \sin (x))) \sqrt{2}}{2} = 0$

Simplify and combine like terms.

Tap for more steps…

Simplify each term.

Tap for more steps…

Multiply $- \cos (x) \cos (x)$ .

Tap for more steps…

Raise $\cos (x)$ to the power of $1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- (\cos (x) \cos (x)) - \cos (x) (- \sin (x)) - \sin (x) \cos (x) - \sin (x) (- \sin (x))) \sqrt{2}}{2} = 0$

Raise $\cos (x)$ to the power of $1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- (\cos (x) \cos (x)) - \cos (x) (- \sin (x)) - \sin (x) \cos (x) - \sin (x) (- \sin (x))) \sqrt{2}}{2} = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sin (x - \frac{π}{4}) + (- {\cos (x)}^{1 + 1} - \cos (x) (- \sin (x)) - \sin (x) \cos (x) - \sin (x) (- \sin (x))) \sqrt{2}}{2} = 0$

Add $1$ and $1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) - \cos (x) (- \sin (x)) - \sin (x) \cos (x) - \sin (x) (- \sin (x))) \sqrt{2}}{2} = 0$

Multiply $- \cos (x) (- \sin (x))$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + 1 \cos (x) \sin (x) - \sin (x) \cos (x) - \sin (x) (- \sin (x))) \sqrt{2}}{2} = 0$

Multiply $\cos (x)$ by $1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + \cos (x) \sin (x) - \sin (x) \cos (x) - \sin (x) (- \sin (x))) \sqrt{2}}{2} = 0$

Multiply $- \sin (x) (- \sin (x))$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + \cos (x) \sin (x) - \sin (x) \cos (x) + 1 \sin (x) \sin (x)) \sqrt{2}}{2} = 0$

Multiply $\sin (x)$ by $1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + \cos (x) \sin (x) - \sin (x) \cos (x) + \sin (x) \sin (x)) \sqrt{2}}{2} = 0$

Raise $\sin (x)$ to the power of $1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + \cos (x) \sin (x) - \sin (x) \cos (x) + \sin (x) \sin (x)) \sqrt{2}}{2} = 0$

Raise $\sin (x)$ to the power of $1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + \cos (x) \sin (x) - \sin (x) \cos (x) + \sin (x) \sin (x)) \sqrt{2}}{2} = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + \cos (x) \sin (x) - \sin (x) \cos (x) + {\sin (x)}^{1 + 1}) \sqrt{2}}{2} = 0$

Add $1$ and $1$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + \cos (x) \sin (x) - \sin (x) \cos (x) + \sin^{2} (x)) \sqrt{2}}{2} = 0$

Reorder the factors of $- \sin (x) \cos (x)$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + \cos (x) \sin (x) - \cos (x) \sin (x) + \sin^{2} (x)) \sqrt{2}}{2} = 0$

Subtract $\cos (x) \sin (x)$ from $\cos (x) \sin (x)$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + 0 + \sin^{2} (x)) \sqrt{2}}{2} = 0$

Add $- \cos^{2} (x)$ and $0$ .

$\frac{2 \sin (x - \frac{π}{4}) + (- \cos^{2} (x) + \sin^{2} (x)) \sqrt{2}}{2} = 0$

Apply the distributive property.

$\frac{2 \sin (x - \frac{π}{4}) - \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2}}{2} = 0$

Factor $- 1$ out of $2 \sin (x - \frac{π}{4}) - \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2}$ .

Tap for more steps…

Move $2 \sin (x - \frac{π}{4})$ .

$\frac{- \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{2} = 0$

Factor $- 1$ out of $- \cos^{2} (x) \sqrt{2}$ .

$\frac{- (\cos^{2} (x) \sqrt{2}) + \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{2} = 0$

Factor $- 1$ out of $\sin^{2} (x) \sqrt{2}$ .

$\frac{- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2}) + 2 \sin (x - \frac{π}{4})}{2} = 0$

Factor $- 1$ out of $2 \sin (x - \frac{π}{4})$ .

$\frac{- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4}))}{2} = 0$

Factor $- 1$ out of $- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2})$ .

$\frac{- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4}))}{2} = 0$

Factor $- 1$ out of $- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4}))$ .

$\frac{- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}))}{2} = 0$

Set $- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}))$ equal to $0$ and solve for $\sin (x - \frac{π}{4})$ .

Tap for more steps…

Set the factor equal to $0$ .

$- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4})) = 0$

Simplify the left side.

Tap for more steps…

Apply the distributive property.

$- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4})) = 0$

Simplify.

Tap for more steps…

Multiply $- (- \sin^{2} (x) \sqrt{2})$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$- \cos^{2} (x) \sqrt{2} + 1 (\sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4})) = 0$

Multiply $\sin^{2} (x)$ by $1$ .

$- \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2} - (- 2 \sin (x - \frac{π}{4})) = 0$

Multiply $- 2$ by $- 1$ .

$- \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4}) = 0$

Move all terms not containing $\sin (x - \frac{π}{4})$ to the right side of the equation.

Tap for more steps…

Add $\cos^{2} (x) \sqrt{2}$ to both sides of the equation.

$\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4}) = \cos^{2} (x) \sqrt{2}$

Subtract $\sin^{2} (x) \sqrt{2}$ from both sides of the equation.

$2 \sin (x - \frac{π}{4}) = \cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}$

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $2 \sin (x - \frac{π}{4}) = \cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}$ by $2$ .

$\frac{2 \sin (x - \frac{π}{4})}{2} = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{2 \sin (x - \frac{π}{4})}{2} = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

Divide $\sin (x - \frac{π}{4})$ by $1$ .

$\sin (x - \frac{π}{4}) = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

The solution is the result of $- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4})) = 0$ .

$\sin (x - \frac{π}{4}) = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

Move $\frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$ to the left side of the equation by subtracting it from both sides.

$\sin (x - \frac{π}{4}) - \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} = 0$

Simplify the left side.

Tap for more steps…

To write $\frac{\sin (x - \frac{π}{4})}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sin (x - \frac{π}{4})}{1} \cdot \frac{2}{2} - \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} = 0$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$\frac{\sin (x - \frac{π}{4}) \cdot 2}{1 \cdot 2} - \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} = 0$

Multiply $2$ by $1$ .

$\frac{\sin (x - \frac{π}{4}) \cdot 2}{2} - \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} = 0$

Combine the numerators over the common denominator.

$\frac{\sin (x - \frac{π}{4}) \cdot 2 - (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2})}{2} = 0$

Simplify the numerator.

Tap for more steps…

Move $2$ to the left of $\sin (x - \frac{π}{4})$ .

$\frac{2 \cdot \sin (x - \frac{π}{4}) - (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2})}{2} = 0$

Apply the distributive property.

$\frac{2 \sin (x - \frac{π}{4}) - (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2})}{2} = 0$

Multiply $- (- \sin^{2} (x) \sqrt{2})$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$\frac{2 \sin (x - \frac{π}{4}) - \cos^{2} (x) \sqrt{2} + 1 (\sin^{2} (x) \sqrt{2})}{2} = 0$

Multiply $\sin^{2} (x)$ by $1$ .

$\frac{2 \sin (x - \frac{π}{4}) - \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2}}{2} = 0$

Factor $- 1$ out of $2 \sin (x - \frac{π}{4}) - \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2}$ .

Tap for more steps…

Move $2 \sin (x - \frac{π}{4})$ .

$\frac{- \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{2} = 0$

Factor $- 1$ out of $- \cos^{2} (x) \sqrt{2}$ .

$\frac{- (\cos^{2} (x) \sqrt{2}) + \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{2} = 0$

Factor $- 1$ out of $\sin^{2} (x) \sqrt{2}$ .

$\frac{- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2}) + 2 \sin (x - \frac{π}{4})}{2} = 0$

Factor $- 1$ out of $2 \sin (x - \frac{π}{4})$ .

$\frac{- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4}))}{2} = 0$

Factor $- 1$ out of $- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2})$ .

$\frac{- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4}))}{2} = 0$

Factor $- 1$ out of $- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4}))$ .

$\frac{- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}))}{2} = 0$

Set $- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}))$ equal to $0$ and solve for $\sin (x - \frac{π}{4})$ .

Tap for more steps…

Set the factor equal to $0$ .

$- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4})) = 0$

Simplify the left side.

Tap for more steps…

Apply the distributive property.

$- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4})) = 0$

Simplify.

Tap for more steps…

Multiply $- (- \sin^{2} (x) \sqrt{2})$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$- \cos^{2} (x) \sqrt{2} + 1 (\sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4})) = 0$

Multiply $\sin^{2} (x)$ by $1$ .

$- \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2} - (- 2 \sin (x - \frac{π}{4})) = 0$

Multiply $- 2$ by $- 1$ .

$- \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4}) = 0$

Move all terms not containing $\sin (x - \frac{π}{4})$ to the right side of the equation.

Tap for more steps…

Add $\cos^{2} (x) \sqrt{2}$ to both sides of the equation.

$\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4}) = \cos^{2} (x) \sqrt{2}$

Subtract $\sin^{2} (x) \sqrt{2}$ from both sides of the equation.

$2 \sin (x - \frac{π}{4}) = \cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}$

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $2 \sin (x - \frac{π}{4}) = \cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}$ by $2$ .

$\frac{2 \sin (x - \frac{π}{4})}{2} = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{2 \sin (x - \frac{π}{4})}{2} = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

Divide $\sin (x - \frac{π}{4})$ by $1$ .

$\sin (x - \frac{π}{4}) = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

The solution is the result of $- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4})) = 0$ .

$\sin (x - \frac{π}{4}) = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

Move $\frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$ to the left side of the equation by subtracting it from both sides.

$\sin (x - \frac{π}{4}) - \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} = 0$

Simplify the left side.

Tap for more steps…

To write $\frac{\sin (x - \frac{π}{4})}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sin (x - \frac{π}{4})}{1} \cdot \frac{2}{2} - \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} = 0$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$\frac{\sin (x - \frac{π}{4}) \cdot 2}{1 \cdot 2} - \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} = 0$

Multiply $2$ by $1$ .

$\frac{\sin (x - \frac{π}{4}) \cdot 2}{2} - \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} = 0$

Combine the numerators over the common denominator.

$\frac{\sin (x - \frac{π}{4}) \cdot 2 - (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2})}{2} = 0$

Simplify the numerator.

Tap for more steps…

Move $2$ to the left of $\sin (x - \frac{π}{4})$ .

$\frac{2 \cdot \sin (x - \frac{π}{4}) - (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2})}{2} = 0$

Apply the distributive property.

$\frac{2 \sin (x - \frac{π}{4}) - (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2})}{2} = 0$

Multiply $- (- \sin^{2} (x) \sqrt{2})$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$\frac{2 \sin (x - \frac{π}{4}) - \cos^{2} (x) \sqrt{2} + 1 (\sin^{2} (x) \sqrt{2})}{2} = 0$

Multiply $\sin^{2} (x)$ by $1$ .

$\frac{2 \sin (x - \frac{π}{4}) - \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2}}{2} = 0$

Factor $- 1$ out of $2 \sin (x - \frac{π}{4}) - \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2}$ .

Tap for more steps…

Move $2 \sin (x - \frac{π}{4})$ .

$\frac{- \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{2} = 0$

Factor $- 1$ out of $- \cos^{2} (x) \sqrt{2}$ .

$\frac{- (\cos^{2} (x) \sqrt{2}) + \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{2} = 0$

Factor $- 1$ out of $\sin^{2} (x) \sqrt{2}$ .

$\frac{- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2}) + 2 \sin (x - \frac{π}{4})}{2} = 0$

Factor $- 1$ out of $2 \sin (x - \frac{π}{4})$ .

$\frac{- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4}))}{2} = 0$

Factor $- 1$ out of $- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2})$ .

$\frac{- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4}))}{2} = 0$

Factor $- 1$ out of $- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4}))$ .

$\frac{- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}))}{2} = 0$

Set $- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}))$ equal to $0$ and solve for $\sin (x - \frac{π}{4})$ .

Tap for more steps…

Set the factor equal to $0$ .

$- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4})) = 0$

Simplify the left side.

Tap for more steps…

Apply the distributive property.

$- (\cos^{2} (x) \sqrt{2}) - (- \sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4})) = 0$

Simplify.

Tap for more steps…

Multiply $- (- \sin^{2} (x) \sqrt{2})$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$- \cos^{2} (x) \sqrt{2} + 1 (\sin^{2} (x) \sqrt{2}) - (- 2 \sin (x - \frac{π}{4})) = 0$

Multiply $\sin^{2} (x)$ by $1$ .

$- \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2} - (- 2 \sin (x - \frac{π}{4})) = 0$

Multiply $- 2$ by $- 1$ .

$- \cos^{2} (x) \sqrt{2} + \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4}) = 0$

Move all terms not containing $\sin (x - \frac{π}{4})$ to the right side of the equation.

Tap for more steps…

Add $\cos^{2} (x) \sqrt{2}$ to both sides of the equation.

$\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4}) = \cos^{2} (x) \sqrt{2}$

Subtract $\sin^{2} (x) \sqrt{2}$ from both sides of the equation.

$2 \sin (x - \frac{π}{4}) = \cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}$

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $2 \sin (x - \frac{π}{4}) = \cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}$ by $2$ .

$\frac{2 \sin (x - \frac{π}{4})}{2} = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{2 \sin (x - \frac{π}{4})}{2} = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

Divide $\sin (x - \frac{π}{4})$ by $1$ .

$\sin (x - \frac{π}{4}) = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

The solution is the result of $- (\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4})) = 0$ .

$\sin (x - \frac{π}{4}) = \frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2}$

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} = \sin (x - \frac{π}{4})$

Move $\sin (x - \frac{π}{4})$ to the left side of the equation by subtracting it from both sides.

$\frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} - \sin (x - \frac{π}{4}) = 0$

Simplify the left side.

Tap for more steps…

To write $\frac{- \sin (x - \frac{π}{4})}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} + \frac{- \sin (x - \frac{π}{4})}{1} \cdot \frac{2}{2} = 0$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$\frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} + \frac{- \sin (x - \frac{π}{4}) \cdot 2}{1 \cdot 2} = 0$

Multiply $2$ by $1$ .

$\frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2}}{2} + \frac{- \sin (x - \frac{π}{4}) \cdot 2}{2} = 0$

Combine the numerators over the common denominator.

$\frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - \sin (x - \frac{π}{4}) \cdot 2}{2} = 0$

Multiply $2$ by $- 1$ .

$\frac{\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4})}{2} = 0$

Multiply both sides of the equation by $2$ .

$\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}) = 0 \cdot (2)$

Remove parentheses.

$\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}) = 0 \cdot (2)$

Multiply $0$ by $2$ .

$\cos^{2} (x) \sqrt{2} - \sin^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}) = 0$

Move all terms not containing $\cos (x)$ to the right side of the equation.

Tap for more steps…

Add $\sin^{2} (x) \sqrt{2}$ to both sides of the equation.

$\cos^{2} (x) \sqrt{2} - 2 \sin (x - \frac{π}{4}) = \sin^{2} (x) \sqrt{2}$

Add $2 \sin (x - \frac{π}{4})$ to both sides of the equation.

$\cos^{2} (x) \sqrt{2} = \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})$

Divide each term by $\sqrt{2}$ and simplify.

Tap for more steps…

Divide each term in $\cos^{2} (x) \sqrt{2} = \sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})$ by $\sqrt{2}$ .

$\frac{\cos^{2} (x) \sqrt{2}}{\sqrt{2}} = \frac{\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{\sqrt{2}}$

Cancel the common factor of $\sqrt{2}$ .

Tap for more steps…

Cancel the common factor.

$\frac{\cos^{2} (x) \sqrt{2}}{\sqrt{2}} = \frac{\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{\sqrt{2}}$

Divide $\cos^{2} (x)$ by $1$ .

$\cos^{2} (x) = \frac{\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{\sqrt{2}}$

Simplify $\frac{\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{\sqrt{2}}$ .

Tap for more steps…

Multiply $\frac{\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos^{2} (x) = \frac{\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Tap for more steps…

Multiply $\frac{\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps…

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{2}$

Evaluate the exponent.

$\cos^{2} (x) = \frac{(\sin^{2} (x) \sqrt{2} + 2 \sin (x - \frac{π}{4})) \sqrt{2}}{2}$

Apply the distributive property.

$\cos^{2} (x) = \frac{\sin^{2} (x) \sqrt{2} \sqrt{2} + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Multiply $\sin^{2} (x) \sqrt{2} \sqrt{2}$ .

Tap for more steps…

Raise $\sqrt{2}$ to the power of $1$ .

$\cos^{2} (x) = \frac{\sin^{2} (x) (\sqrt{2} \sqrt{2}) + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$\cos^{2} (x) = \frac{\sin^{2} (x) (\sqrt{2} \sqrt{2}) + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos^{2} (x) = \frac{\sin^{2} (x) {\sqrt{2}}^{1 + 1} + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Add $1$ and $1$ .

$\cos^{2} (x) = \frac{\sin^{2} (x) {\sqrt{2}}^{2} + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Simplify each term.

Tap for more steps…

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps…

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\cos^{2} (x) = \frac{\sin^{2} (x) {(2^{\frac{1}{2}})}^{2} + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos^{2} (x) = \frac{\sin^{2} (x) \cdot 2^{\frac{1}{2} \cdot 2} + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Combine $\frac{1}{2}$ and $2$ .

$\cos^{2} (x) = \frac{\sin^{2} (x) \cdot 2^{\frac{2}{2}} + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\cos^{2} (x) = \frac{\sin^{2} (x) \cdot 2^{\frac{2}{2}} + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Divide $1$ by $1$ .

$\cos^{2} (x) = \frac{\sin^{2} (x) \cdot 2 + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Evaluate the exponent.

$\cos^{2} (x) = \frac{\sin^{2} (x) \cdot 2 + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Move $2$ to the left of $\sin^{2} (x)$ .

$\cos^{2} (x) = \frac{2 \sin^{2} (x) + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Cancel the common factor of $2 \sin^{2} (x) + 2 \sin (x - \frac{π}{4}) \sqrt{2}$ and $2$ .

Tap for more steps…

Factor $2$ out of $2 \sin^{2} (x)$ .

$\cos^{2} (x) = \frac{2 (\sin^{2} (x)) + 2 \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Factor $2$ out of $2 \sin (x - \frac{π}{4}) \sqrt{2}$ .

$\cos^{2} (x) = \frac{2 (\sin^{2} (x)) + 2 (\sin (x - \frac{π}{4}) \sqrt{2})}{2}$

Factor $2$ out of $2 \sin^{2} (x) + 2 (\sin (x - \frac{π}{4}) \sqrt{2})$ .

$\cos^{2} (x) = \frac{2 (\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2})}{2}$

Cancel the common factors.

Tap for more steps…

Factor $2$ out of $2$ .

$\cos^{2} (x) = \frac{2 (\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2})}{2 (1)}$

Cancel the common factor.

$\cos^{2} (x) = \frac{2 (\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2})}{2 \cdot 1}$

Rewrite the expression.

$\cos^{2} (x) = \frac{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}{1}$

Divide $\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}$ by $1$ .

$\cos^{2} (x) = \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}$

Take the $square$ root of both sides of the $equation$ to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}$

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps…

First, use the positive value of the $\pm$ to find the first solution.

$\cos (x) = \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}$

Next, use the negative value of the $\pm$ to find the second solution.

$\cos (x) = - \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}$

The complete solution is the result of both the positive and negative portions of the solution.

$\cos (x) = \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}, - \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}$

Set up each of the solutions to solve for $x$ .

$\cos (x) = \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}$

$\cos (x) = - \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}$

Set up the equation to solve for $x$ .

$\cos (x) = \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}$

Solve the equation for $x$ .

Tap for more steps…

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}} = \cos (x)$

To remove the radical on the left side of the equation, square both sides of the equation.

${(\sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}})}^{2} = \cos^{2} (x)$

Simplify the left side of the equation.

$\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = \cos^{2} (x)$

Solve for $x$ .

Tap for more steps…

Move $\cos^{2} (x)$ to the left side of the equation by subtracting it from both sides.

$\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - \cos^{2} (x) = 0$

Replace $\cos^{2} (x)$ with $1 - \sin^{2} (x)$ .

$\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - (1 - \sin^{2} (x)) = 0$

Simplify $- (1 - \sin^{2} (x))$ .

$\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - 1 + \sin^{2} (x) = 0$

Replace the $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$1 - \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - 1 + \sin^{2} (x) = 0$

Combine the opposite terms in $1 - \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - 1 + \sin^{2} (x)$ .

Tap for more steps…

Subtract $1$ from $1$ .

$- \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} + 0 + \sin^{2} (x) = 0$

Add $- \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}$ and $0$ .

$- \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} + \sin^{2} (x) = 0$

Reorder the polynomial.

$- \cos^{2} (x) + \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 0$

Replace $\cos^{2} (x)$ with $1 - \sin^{2} (x)$ .

$- (1 - \sin^{2} (x)) + \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 0$

Simplify $- (1 - \sin^{2} (x))$ .

$- 1 + \sin^{2} (x) + \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 0$

Add $\sin^{2} (x)$ and $\sin^{2} (x)$ .

$- 1 + 2 \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 0$

Move all terms not containing $\sin (x)$ to the right side of the equation.

Tap for more steps…

Add $1$ to both sides of the equation.

$2 \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 1$

Subtract $\sin (x - \frac{π}{4}) \sqrt{2}$ from both sides of the equation.

$2 \sin^{2} (x) = 1 - \sin (x - \frac{π}{4}) \sqrt{2}$

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $2 \sin^{2} (x) = 1 - \sin (x - \frac{π}{4}) \sqrt{2}$ by $2$ .

$\frac{2 \sin^{2} (x)}{2} = \frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{2 \sin^{2} (x)}{2} = \frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Divide $\sin^{2} (x)$ by $1$ .

$\sin^{2} (x) = \frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Take the $square$ root of both sides of the $equation$ to eliminate the exponent on the left side.

$\sin (x) = \pm \sqrt{\frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}}$

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps…

Simplify the right side of the equation.

Tap for more steps…

Rewrite $\sqrt{\frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}}$ as $\frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}}$

Multiply $\frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Tap for more steps…

Multiply $\frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps…

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2}$

Evaluate the exponent.

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2}$

Combine using the product rule for radicals.

$\sin (x) = \pm \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps…

First, use the positive value of the $\pm$ to find the first solution.

$\sin (x) = \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$\sin (x) = - \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$\sin (x) = \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}, - \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

$\sin (x) = - \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Set up the equation to solve for $x$ .

$\sin (x) = \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Solve the equation for $x$ .

Tap for more steps…

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Simplify the numerator.

Tap for more steps…

To write $\frac{x}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\sqrt{(1 - \sin (\frac{x}{1} \cdot \frac{4}{4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$\frac{\sqrt{(1 - \sin (\frac{x \cdot 4}{1 \cdot 4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Multiply $4$ by $1$ .

$\frac{\sqrt{(1 - \sin (\frac{x \cdot 4}{4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Combine the numerators over the common denominator.

$\frac{\sqrt{(1 - \sin (\frac{x \cdot 4 - π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Move $4$ to the left of $x$ .

$\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Multiply both sides of the equation by $2$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = \sin (x) \cdot (2)$

Move $2$ to the left of $\sin (x)$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 2 \sin (x)$

Move all terms containing $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ to the left side of the equation.

Tap for more steps…

Subtract $2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ from both sides of the equation.

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} - 2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

Subtract $2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ from $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ .

$- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

Multiply each term in $- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$ by $- 1$

Tap for more steps…

Multiply each term in $- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$ by $- 1$ .

$(- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}) \cdot - 1 = 0 \cdot - 1$

Multiply $- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} \cdot - 1$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$1 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0 \cdot - 1$

Multiply $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ by $1$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0 \cdot - 1$

Multiply $0$ by $- 1$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

To remove the radical on the left side of the equation, square both sides of the equation.

${(\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2})}^{2} = {(0)}^{2}$

Simplify each side of the equation.

Tap for more steps…

Simplify the left side of the equation.

$(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = {(0)}^{2}$

Raising $0$ to any positive power yields $0$ .

$(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = 0$

Solve for $\sin (\frac{4 x - π}{4})$ .

Tap for more steps…

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = 0$ by $2$ .

$\frac{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}{2} = \frac{0}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}{2} = \frac{0}{2}$

Divide $1 - \sin (\frac{4 x - π}{4}) \sqrt{2}$ by $1$ .

$1 - \sin (\frac{4 x - π}{4}) \sqrt{2} = \frac{0}{2}$

Divide $0$ by $2$ .

$1 - \sin (\frac{4 x - π}{4}) \sqrt{2} = 0$

Subtract $1$ from both sides of the equation.

$- \sin (\frac{4 x - π}{4}) \sqrt{2} = - 1$

Divide each term by $- \sqrt{2}$ and simplify.

Tap for more steps…

Divide each term in $- \sin (\frac{4 x - π}{4}) \sqrt{2} = - 1$ by $- \sqrt{2}$ .

$\frac{- \sin (\frac{4 x - π}{4}) \sqrt{2}}{- \sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Simplify the left side of the equation by multiplying out all the terms.

Tap for more steps…

Dividing two negative values results in a positive value.

$\frac{\sin (\frac{4 x - π}{4}) \sqrt{2}}{\sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Cancel the common factor of $\sqrt{2}$ .

Tap for more steps…

Cancel the common factor.

$\frac{\sin (\frac{4 x - π}{4}) \sqrt{2}}{\sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Divide $\sin (\frac{4 x - π}{4})$ by $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{- 1}{- \sqrt{2}}$

Simplify $\frac{- 1}{- \sqrt{2}}$ .

Tap for more steps…

Dividing two negative values results in a positive value.

$\sin (\frac{4 x - π}{4}) = \frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Tap for more steps…

Multiply $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps…

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2}$

Evaluate the exponent.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2}$

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$\frac{4 x - π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Simplify the left side.

Tap for more steps…

Split the fraction $\frac{4 x - π}{4}$ into two fractions.

$\frac{4 x}{4} + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Simplify each term.

Tap for more steps…

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

$\frac{4 x}{4} + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Divide $x$ by $1$ .

$x + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Move the negative in front of the fraction.

$x - \frac{π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

The exact value of $\sin^{-1} (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x - \frac{π}{4} = \frac{π}{4}$

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps…

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{π}{4} + \frac{π}{4}$

Simplify the right side of the equation.

Tap for more steps…

Combine the numerators over the common denominator.

$x = \frac{π + π}{4}$

Add $π$ and $π$ .

$x = \frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

Tap for more steps…

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{4}$

Cancel the common factors.

Tap for more steps…

Factor $2$ out of $4$ .

$x = \frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{π}{2}$

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x - \frac{π}{4} = π - \frac{π}{4}$

Simplify the expression to find the second solution.

Tap for more steps…

Simplify $π - \frac{π}{4}$ .

Tap for more steps…

To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x - \frac{π}{4} = \frac{π}{1} \cdot \frac{4}{4} - \frac{π}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$x - \frac{π}{4} = \frac{π \cdot 4}{1 \cdot 4} - \frac{π}{4}$

Multiply $4$ by $1$ .

$x - \frac{π}{4} = \frac{π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$x - \frac{π}{4} = \frac{π \cdot 4 - π}{4}$

Simplify the numerator.

Tap for more steps…

Move $4$ to the left of $π$ .

$x - \frac{π}{4} = \frac{4 \cdot π - π}{4}$

Subtract $π$ from $4 π$ .

$x - \frac{π}{4} = \frac{3 π}{4}$

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps…

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{3 π}{4} + \frac{π}{4}$

Simplify the right side of the equation.

Tap for more steps…

Combine the numerators over the common denominator.

$x = \frac{3 π + π}{4}$

Add $3 π$ and $π$ .

$x = \frac{4 π}{4}$

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

$x = \frac{4 π}{4}$

Divide $π$ by $1$ .

$x = π$

Find the period.

Tap for more steps…

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

Solve the equation.

Tap for more steps…

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Cancel the common factor of $π$ .

Tap for more steps…

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Rewrite the expression.

$2 \cdot 4$

Multiply $2$ by $4$ .

$8$

The period of the $\sin (\frac{4 x - π}{4})$ function is $8$ so values will repeat every $8$ radians in both directions.

$x = \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

Set up the equation to solve for $x$ .

$\sin (x) = - \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Solve the equation for $x$ .

Tap for more steps…

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Simplify the numerator.

Tap for more steps…

To write $\frac{x}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{\sqrt{(1 - \sin (\frac{x}{1} \cdot \frac{4}{4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$- \frac{\sqrt{(1 - \sin (\frac{x \cdot 4}{1 \cdot 4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Multiply $4$ by $1$ .

$- \frac{\sqrt{(1 - \sin (\frac{x \cdot 4}{4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Combine the numerators over the common denominator.

$- \frac{\sqrt{(1 - \sin (\frac{x \cdot 4 - π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Move $4$ to the left of $x$ .

$- \frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Multiply both sides of the equation by $- 1$ .

$- 1 \cdot (- \frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2}) = - 1 \cdot \sin (x)$

Simplify both sides of the equation.

Tap for more steps…

Multiply $- 1 (- \frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2})$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$1 (\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2}) = - 1 \cdot \sin (x)$

Multiply $\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2}$ by $1$ .

$\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2} = - 1 \cdot \sin (x)$

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

$\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2} = - \sin (x)$

Multiply both sides of the equation by $2$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = - \sin (x) \cdot 2$

Multiply $2$ by $- 1$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = - 2 \sin (x)$

Move all terms containing $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ to the left side of the equation.

Tap for more steps…

Add $- 2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ to both sides of the equation.

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} + 2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

Add $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ and $2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ .

$3 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

Divide each term by $3$ and simplify.

Tap for more steps…

Divide each term in $3 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$ by $3$ .

$\frac{3 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{3} = \frac{0}{3}$

Cancel the common factor of $3$ .

Tap for more steps…

Cancel the common factor.

$\frac{3 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{3} = \frac{0}{3}$

Divide $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ by $1$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = \frac{0}{3}$

Divide $0$ by $3$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

To remove the radical on the left side of the equation, square both sides of the equation.

${(\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2})}^{2} = {(0)}^{2}$

Simplify each side of the equation.

Tap for more steps…

Simplify the left side of the equation.

$(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = {(0)}^{2}$

Raising $0$ to any positive power yields $0$ .

$(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = 0$

Solve for $\sin (\frac{4 x - π}{4})$ .

Tap for more steps…

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = 0$ by $2$ .

$\frac{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}{2} = \frac{0}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}{2} = \frac{0}{2}$

Divide $1 - \sin (\frac{4 x - π}{4}) \sqrt{2}$ by $1$ .

$1 - \sin (\frac{4 x - π}{4}) \sqrt{2} = \frac{0}{2}$

Divide $0$ by $2$ .

$1 - \sin (\frac{4 x - π}{4}) \sqrt{2} = 0$

Subtract $1$ from both sides of the equation.

$- \sin (\frac{4 x - π}{4}) \sqrt{2} = - 1$

Divide each term by $- \sqrt{2}$ and simplify.

Tap for more steps…

Divide each term in $- \sin (\frac{4 x - π}{4}) \sqrt{2} = - 1$ by $- \sqrt{2}$ .

$\frac{- \sin (\frac{4 x - π}{4}) \sqrt{2}}{- \sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Simplify the left side of the equation by multiplying out all the terms.

Tap for more steps…

Dividing two negative values results in a positive value.

$\frac{\sin (\frac{4 x - π}{4}) \sqrt{2}}{\sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Cancel the common factor of $\sqrt{2}$ .

Tap for more steps…

Cancel the common factor.

$\frac{\sin (\frac{4 x - π}{4}) \sqrt{2}}{\sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Divide $\sin (\frac{4 x - π}{4})$ by $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{- 1}{- \sqrt{2}}$

Simplify $\frac{- 1}{- \sqrt{2}}$ .

Tap for more steps…

Dividing two negative values results in a positive value.

$\sin (\frac{4 x - π}{4}) = \frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Tap for more steps…

Multiply $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps…

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2}$

Evaluate the exponent.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2}$

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$\frac{4 x - π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Simplify the left side.

Tap for more steps…

Split the fraction $\frac{4 x - π}{4}$ into two fractions.

$\frac{4 x}{4} + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Simplify each term.

Tap for more steps…

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

$\frac{4 x}{4} + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Divide $x$ by $1$ .

$x + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Move the negative in front of the fraction.

$x - \frac{π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

The exact value of $\sin^{-1} (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x - \frac{π}{4} = \frac{π}{4}$

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps…

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{π}{4} + \frac{π}{4}$

Simplify the right side of the equation.

Tap for more steps…

Combine the numerators over the common denominator.

$x = \frac{π + π}{4}$

Add $π$ and $π$ .

$x = \frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

Tap for more steps…

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{4}$

Cancel the common factors.

Tap for more steps…

Factor $2$ out of $4$ .

$x = \frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{π}{2}$

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x - \frac{π}{4} = π - \frac{π}{4}$

Simplify the expression to find the second solution.

Tap for more steps…

Simplify $π - \frac{π}{4}$ .

Tap for more steps…

To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x - \frac{π}{4} = \frac{π}{1} \cdot \frac{4}{4} - \frac{π}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$x - \frac{π}{4} = \frac{π \cdot 4}{1 \cdot 4} - \frac{π}{4}$

Multiply $4$ by $1$ .

$x - \frac{π}{4} = \frac{π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$x - \frac{π}{4} = \frac{π \cdot 4 - π}{4}$

Simplify the numerator.

Tap for more steps…

Move $4$ to the left of $π$ .

$x - \frac{π}{4} = \frac{4 \cdot π - π}{4}$

Subtract $π$ from $4 π$ .

$x - \frac{π}{4} = \frac{3 π}{4}$

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps…

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{3 π}{4} + \frac{π}{4}$

Simplify the right side of the equation.

Tap for more steps…

Combine the numerators over the common denominator.

$x = \frac{3 π + π}{4}$

Add $3 π$ and $π$ .

$x = \frac{4 π}{4}$

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

$x = \frac{4 π}{4}$

Divide $π$ by $1$ .

$x = π$

Find the period.

Tap for more steps…

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

Solve the equation.

Tap for more steps…

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Cancel the common factor of $π$ .

Tap for more steps…

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Rewrite the expression.

$2 \cdot 4$

Multiply $2$ by $4$ .

$8$

The period of the $\sin (\frac{4 x - π}{4})$ function is $8$ so values will repeat every $8$ radians in both directions.

$x = \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

List all of the results found in the previous steps.

$x = \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

The complete solution is the set of all solutions.

$x = \frac{π}{2} + 8 n, π + 8 n, \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

Set up the equation to solve for $x$ .

$\cos (x) = - \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}}$

Solve the equation for $x$ .

Tap for more steps…

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}} = \cos (x)$

To remove the radical on the left side of the equation, square both sides of the equation.

${(- \sqrt{\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}})}^{2} = \cos^{2} (x)$

Simplify the left side of the equation.

${(- 1)}^{2} (\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}) = \cos^{2} (x)$

Solve for $x$ .

Tap for more steps…

Move $\cos^{2} (x)$ to the left side of the equation by subtracting it from both sides.

${(- 1)}^{2} (\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}) - \cos^{2} (x) = 0$

Simplify each term.

Tap for more steps…

Raise $- 1$ to the power of $2$ .

$1 (\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}) - \cos^{2} (x) = 0$

Multiply $\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}$ by $1$ .

$\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - \cos^{2} (x) = 0$

Replace $\cos^{2} (x)$ with $1 - \sin^{2} (x)$ .

$\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - (1 - \sin^{2} (x)) = 0$

Simplify $- (1 - \sin^{2} (x))$ .

$\sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - 1 + \sin^{2} (x) = 0$

Replace the $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$1 - \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - 1 + \sin^{2} (x) = 0$

Combine the opposite terms in $1 - \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} - 1 + \sin^{2} (x)$ .

Tap for more steps…

Subtract $1$ from $1$ .

$- \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} + 0 + \sin^{2} (x) = 0$

Add $- \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2}$ and $0$ .

$- \cos^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} + \sin^{2} (x) = 0$

Reorder the polynomial.

$- \cos^{2} (x) + \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 0$

Replace $\cos^{2} (x)$ with $1 - \sin^{2} (x)$ .

$- (1 - \sin^{2} (x)) + \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 0$

Simplify $- (1 - \sin^{2} (x))$ .

$- 1 + \sin^{2} (x) + \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 0$

Add $\sin^{2} (x)$ and $\sin^{2} (x)$ .

$- 1 + 2 \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 0$

Move all terms not containing $\sin (x)$ to the right side of the equation.

Tap for more steps…

Add $1$ to both sides of the equation.

$2 \sin^{2} (x) + \sin (x - \frac{π}{4}) \sqrt{2} = 1$

Subtract $\sin (x - \frac{π}{4}) \sqrt{2}$ from both sides of the equation.

$2 \sin^{2} (x) = 1 - \sin (x - \frac{π}{4}) \sqrt{2}$

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $2 \sin^{2} (x) = 1 - \sin (x - \frac{π}{4}) \sqrt{2}$ by $2$ .

$\frac{2 \sin^{2} (x)}{2} = \frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{2 \sin^{2} (x)}{2} = \frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Divide $\sin^{2} (x)$ by $1$ .

$\sin^{2} (x) = \frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}$

Take the $square$ root of both sides of the $equation$ to eliminate the exponent on the left side.

$\sin (x) = \pm \sqrt{\frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}}$

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps…

Simplify the right side of the equation.

Tap for more steps…

Rewrite $\sqrt{\frac{1 - \sin (x - \frac{π}{4}) \sqrt{2}}{2}}$ as $\frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}}$

Multiply $\frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Tap for more steps…

Multiply $\frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}}}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps…

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2}$

Evaluate the exponent.

$\sin (x) = \pm \frac{\sqrt{1 - \sin (x - \frac{π}{4}) \sqrt{2}} \sqrt{2}}{2}$

Combine using the product rule for radicals.

$\sin (x) = \pm \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps…

First, use the positive value of the $\pm$ to find the first solution.

$\sin (x) = \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$\sin (x) = - \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$\sin (x) = \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}, - \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

$\sin (x) = - \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Set up the equation to solve for $x$ .

$\sin (x) = \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Solve the equation for $x$ .

Tap for more steps…

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Simplify the numerator.

Tap for more steps…

To write $\frac{x}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\sqrt{(1 - \sin (\frac{x}{1} \cdot \frac{4}{4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$\frac{\sqrt{(1 - \sin (\frac{x \cdot 4}{1 \cdot 4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Multiply $4$ by $1$ .

$\frac{\sqrt{(1 - \sin (\frac{x \cdot 4}{4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Combine the numerators over the common denominator.

$\frac{\sqrt{(1 - \sin (\frac{x \cdot 4 - π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Move $4$ to the left of $x$ .

$\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Multiply both sides of the equation by $2$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = \sin (x) \cdot (2)$

Move $2$ to the left of $\sin (x)$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 2 \sin (x)$

Move all terms containing $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ to the left side of the equation.

Tap for more steps…

Subtract $2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ from both sides of the equation.

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} - 2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

Subtract $2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ from $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ .

$- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

Multiply each term in $- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$ by $- 1$

Tap for more steps…

Multiply each term in $- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$ by $- 1$ .

$(- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}) \cdot - 1 = 0 \cdot - 1$

Multiply $- \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} \cdot - 1$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$1 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0 \cdot - 1$

Multiply $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ by $1$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0 \cdot - 1$

Multiply $0$ by $- 1$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

To remove the radical on the left side of the equation, square both sides of the equation.

${(\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2})}^{2} = {(0)}^{2}$

Simplify each side of the equation.

Tap for more steps…

Simplify the left side of the equation.

$(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = {(0)}^{2}$

Raising $0$ to any positive power yields $0$ .

$(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = 0$

Solve for $\sin (\frac{4 x - π}{4})$ .

Tap for more steps…

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = 0$ by $2$ .

$\frac{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}{2} = \frac{0}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}{2} = \frac{0}{2}$

Divide $1 - \sin (\frac{4 x - π}{4}) \sqrt{2}$ by $1$ .

$1 - \sin (\frac{4 x - π}{4}) \sqrt{2} = \frac{0}{2}$

Divide $0$ by $2$ .

$1 - \sin (\frac{4 x - π}{4}) \sqrt{2} = 0$

Subtract $1$ from both sides of the equation.

$- \sin (\frac{4 x - π}{4}) \sqrt{2} = - 1$

Divide each term by $- \sqrt{2}$ and simplify.

Tap for more steps…

Divide each term in $- \sin (\frac{4 x - π}{4}) \sqrt{2} = - 1$ by $- \sqrt{2}$ .

$\frac{- \sin (\frac{4 x - π}{4}) \sqrt{2}}{- \sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Simplify the left side of the equation by multiplying out all the terms.

Tap for more steps…

Dividing two negative values results in a positive value.

$\frac{\sin (\frac{4 x - π}{4}) \sqrt{2}}{\sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Cancel the common factor of $\sqrt{2}$ .

Tap for more steps…

Cancel the common factor.

$\frac{\sin (\frac{4 x - π}{4}) \sqrt{2}}{\sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Divide $\sin (\frac{4 x - π}{4})$ by $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{- 1}{- \sqrt{2}}$

Simplify $\frac{- 1}{- \sqrt{2}}$ .

Tap for more steps…

Dividing two negative values results in a positive value.

$\sin (\frac{4 x - π}{4}) = \frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Tap for more steps…

Multiply $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps…

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2}$

Evaluate the exponent.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2}$

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$\frac{4 x - π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Simplify the left side.

Tap for more steps…

Split the fraction $\frac{4 x - π}{4}$ into two fractions.

$\frac{4 x}{4} + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Simplify each term.

Tap for more steps…

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

$\frac{4 x}{4} + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Divide $x$ by $1$ .

$x + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Move the negative in front of the fraction.

$x - \frac{π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

The exact value of $\sin^{-1} (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x - \frac{π}{4} = \frac{π}{4}$

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps…

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{π}{4} + \frac{π}{4}$

Simplify the right side of the equation.

Tap for more steps…

Combine the numerators over the common denominator.

$x = \frac{π + π}{4}$

Add $π$ and $π$ .

$x = \frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

Tap for more steps…

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{4}$

Cancel the common factors.

Tap for more steps…

Factor $2$ out of $4$ .

$x = \frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{π}{2}$

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x - \frac{π}{4} = π - \frac{π}{4}$

Simplify the expression to find the second solution.

Tap for more steps…

Simplify $π - \frac{π}{4}$ .

Tap for more steps…

To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x - \frac{π}{4} = \frac{π}{1} \cdot \frac{4}{4} - \frac{π}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$x - \frac{π}{4} = \frac{π \cdot 4}{1 \cdot 4} - \frac{π}{4}$

Multiply $4$ by $1$ .

$x - \frac{π}{4} = \frac{π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$x - \frac{π}{4} = \frac{π \cdot 4 - π}{4}$

Simplify the numerator.

Tap for more steps…

Move $4$ to the left of $π$ .

$x - \frac{π}{4} = \frac{4 \cdot π - π}{4}$

Subtract $π$ from $4 π$ .

$x - \frac{π}{4} = \frac{3 π}{4}$

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps…

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{3 π}{4} + \frac{π}{4}$

Simplify the right side of the equation.

Tap for more steps…

Combine the numerators over the common denominator.

$x = \frac{3 π + π}{4}$

Add $3 π$ and $π$ .

$x = \frac{4 π}{4}$

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

$x = \frac{4 π}{4}$

Divide $π$ by $1$ .

$x = π$

Find the period.

Tap for more steps…

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

Solve the equation.

Tap for more steps…

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Cancel the common factor of $π$ .

Tap for more steps…

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Rewrite the expression.

$2 \cdot 4$

Multiply $2$ by $4$ .

$8$

The period of the $\sin (\frac{4 x - π}{4})$ function is $8$ so values will repeat every $8$ radians in both directions.

$x = \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

Set up the equation to solve for $x$ .

$\sin (x) = - \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2}$

Solve the equation for $x$ .

Tap for more steps…

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \frac{\sqrt{(1 - \sin (x - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Simplify the numerator.

Tap for more steps…

To write $\frac{x}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{\sqrt{(1 - \sin (\frac{x}{1} \cdot \frac{4}{4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$- \frac{\sqrt{(1 - \sin (\frac{x \cdot 4}{1 \cdot 4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Multiply $4$ by $1$ .

$- \frac{\sqrt{(1 - \sin (\frac{x \cdot 4}{4} - \frac{π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Combine the numerators over the common denominator.

$- \frac{\sqrt{(1 - \sin (\frac{x \cdot 4 - π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Move $4$ to the left of $x$ .

$- \frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2} = \sin (x)$

Multiply both sides of the equation by $- 1$ .

$- 1 \cdot (- \frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2}) = - 1 \cdot \sin (x)$

Simplify both sides of the equation.

Tap for more steps…

Multiply $- 1 (- \frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2})$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$1 (\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2}) = - 1 \cdot \sin (x)$

Multiply $\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2}$ by $1$ .

$\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2} = - 1 \cdot \sin (x)$

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

$\frac{\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{2} = - \sin (x)$

Multiply both sides of the equation by $2$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = - \sin (x) \cdot 2$

Multiply $2$ by $- 1$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = - 2 \sin (x)$

Move all terms containing $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ to the left side of the equation.

Tap for more steps…

Add $- 2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ to both sides of the equation.

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} + 2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

Add $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ and $2 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ .

$3 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

Divide each term by $3$ and simplify.

Tap for more steps…

Divide each term in $3 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$ by $3$ .

$\frac{3 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{3} = \frac{0}{3}$

Cancel the common factor of $3$ .

Tap for more steps…

Cancel the common factor.

$\frac{3 \sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}}{3} = \frac{0}{3}$

Divide $\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}$ by $1$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = \frac{0}{3}$

Divide $0$ by $3$ .

$\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2} = 0$

To remove the radical on the left side of the equation, square both sides of the equation.

${(\sqrt{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2})}^{2} = {(0)}^{2}$

Simplify each side of the equation.

Tap for more steps…

Simplify the left side of the equation.

$(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = {(0)}^{2}$

Raising $0$ to any positive power yields $0$ .

$(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = 0$

Solve for $\sin (\frac{4 x - π}{4})$ .

Tap for more steps…

Divide each term by $2$ and simplify.

Tap for more steps…

Divide each term in $(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2 = 0$ by $2$ .

$\frac{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}{2} = \frac{0}{2}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\frac{(1 - \sin (\frac{4 x - π}{4}) \sqrt{2}) \cdot 2}{2} = \frac{0}{2}$

Divide $1 - \sin (\frac{4 x - π}{4}) \sqrt{2}$ by $1$ .

$1 - \sin (\frac{4 x - π}{4}) \sqrt{2} = \frac{0}{2}$

Divide $0$ by $2$ .

$1 - \sin (\frac{4 x - π}{4}) \sqrt{2} = 0$

Subtract $1$ from both sides of the equation.

$- \sin (\frac{4 x - π}{4}) \sqrt{2} = - 1$

Divide each term by $- \sqrt{2}$ and simplify.

Tap for more steps…

Divide each term in $- \sin (\frac{4 x - π}{4}) \sqrt{2} = - 1$ by $- \sqrt{2}$ .

$\frac{- \sin (\frac{4 x - π}{4}) \sqrt{2}}{- \sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Simplify the left side of the equation by multiplying out all the terms.

Tap for more steps…

Dividing two negative values results in a positive value.

$\frac{\sin (\frac{4 x - π}{4}) \sqrt{2}}{\sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Cancel the common factor of $\sqrt{2}$ .

Tap for more steps…

Cancel the common factor.

$\frac{\sin (\frac{4 x - π}{4}) \sqrt{2}}{\sqrt{2}} = \frac{- 1}{- \sqrt{2}}$

Divide $\sin (\frac{4 x - π}{4})$ by $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{- 1}{- \sqrt{2}}$

Simplify $\frac{- 1}{- \sqrt{2}}$ .

Tap for more steps…

Dividing two negative values results in a positive value.

$\sin (\frac{4 x - π}{4}) = \frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Tap for more steps…

Multiply $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps…

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2}$

Evaluate the exponent.

$\sin (\frac{4 x - π}{4}) = \frac{\sqrt{2}}{2}$

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$\frac{4 x - π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Simplify the left side.

Tap for more steps…

Split the fraction $\frac{4 x - π}{4}$ into two fractions.

$\frac{4 x}{4} + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Simplify each term.

Tap for more steps…

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

$\frac{4 x}{4} + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Divide $x$ by $1$ .

$x + \frac{- π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Move the negative in front of the fraction.

$x - \frac{π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

The exact value of $\sin^{-1} (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x - \frac{π}{4} = \frac{π}{4}$

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps…

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{π}{4} + \frac{π}{4}$

Simplify the right side of the equation.

Tap for more steps…

Combine the numerators over the common denominator.

$x = \frac{π + π}{4}$

Add $π$ and $π$ .

$x = \frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

Tap for more steps…

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{4}$

Cancel the common factors.

Tap for more steps…

Factor $2$ out of $4$ .

$x = \frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{π}{2}$

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x - \frac{π}{4} = π - \frac{π}{4}$

Simplify the expression to find the second solution.

Tap for more steps…

Simplify $π - \frac{π}{4}$ .

Tap for more steps…

To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x - \frac{π}{4} = \frac{π}{1} \cdot \frac{4}{4} - \frac{π}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Combine.

$x - \frac{π}{4} = \frac{π \cdot 4}{1 \cdot 4} - \frac{π}{4}$

Multiply $4$ by $1$ .

$x - \frac{π}{4} = \frac{π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$x - \frac{π}{4} = \frac{π \cdot 4 - π}{4}$

Simplify the numerator.

Tap for more steps…

Move $4$ to the left of $π$ .

$x - \frac{π}{4} = \frac{4 \cdot π - π}{4}$

Subtract $π$ from $4 π$ .

$x - \frac{π}{4} = \frac{3 π}{4}$

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps…

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{3 π}{4} + \frac{π}{4}$

Simplify the right side of the equation.

Tap for more steps…

Combine the numerators over the common denominator.

$x = \frac{3 π + π}{4}$

Add $3 π$ and $π$ .

$x = \frac{4 π}{4}$

Cancel the common factor of $4$ .

Tap for more steps…

Cancel the common factor.

$x = \frac{4 π}{4}$

Divide $π$ by $1$ .

$x = π$

Find the period.

Tap for more steps…

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

Solve the equation.

Tap for more steps…

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Cancel the common factor of $π$ .

Tap for more steps…

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Rewrite the expression.

$2 \cdot 4$

Multiply $2$ by $4$ .

$8$

The period of the $\sin (\frac{4 x - π}{4})$ function is $8$ so values will repeat every $8$ radians in both directions.

$x = \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

List all of the results found in the previous steps.

$x = \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

The complete solution is the set of all solutions.

$x = \frac{π}{2} + 8 n, π + 8 n, \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

List all of the results found in the previous steps.

$x = \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

The complete solution is the set of all solutions.

$x = \frac{π}{2} + 8 n, π + 8 n, \frac{π}{2} + 8 n, π + 8 n, \frac{π}{2} + 8 n, π + 8 n, \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

Consolidate the answers.

Tap for more steps…

Consolidate $\frac{π}{2} + 8 n$ and $\frac{π}{2} + 8 n$ to $\frac{π}{2} + 8 n$ .

$x = π + 8 n, π + 8 n, \frac{π}{2} + 8 n, π + 8 n, \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

Consolidate $π + 8 n$ and $π + 8 n$ to $π + 8 n$ .

$x = \frac{π}{2} + 8 n, π + 8 n, \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

Consolidate $\frac{π}{2} + 8 n$ and $\frac{π}{2} + 8 n$ to $\frac{π}{2} + 8 n$ .

$x = \frac{π}{2} + 8 n, π + 8 n, π + 8 n$ , for any integer $n$

Consolidate $π + 8 n$ and $π + 8 n$ to $π + 8 n$ .

$x = \frac{π}{2} + 8 n, π + 8 n$ , for any integer $n$

Solve for x cos(x)^4-sin(x)^4=cos(2x)

Download our
App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our
App from the store