Find the Inverse cos(4x)-sin(4x)

$\cos (4 x) - \sin (4 x)$

Interchange the variables.

$x = \cos (4 y) - \sin (4 y)$

Solve for $y$ .

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Rewrite the equation as $\cos (4 y) - \sin (4 y) = x$ .

$\cos (4 y) - \sin (4 y) = x$

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 $θ$ .

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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 $R$ .

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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.

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

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

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

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

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

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Cancel the common factor.

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

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

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

Simplify $\frac{x}{\sqrt{2}}$ .

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Evaluate the exponent.

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

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

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

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

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

Divide each term by $4$ and simplify.

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

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Divide $y$ by $1$ .

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

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

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Simplify the numerator.

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

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

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

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Combine.

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

Multiply $4$ by $1$ .

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

Combine the numerators over the common denominator.

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

Move $4$ to the left of $\sin^{-1} (\frac{x \sqrt{2}}{2})$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $4$ by $4$ .

$y = \frac{4 \sin^{-1} (\frac{x \sqrt{2}}{2}) + π}{16}$

Solve for $y$ and replace with $f^{- 1} (x)$ .

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Replace the $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{4 \sin^{-1} (\frac{x \sqrt{2}}{2}) + π}{16}$

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

$\frac{4 \sin^{-1} (\frac{(\cos (4 x) - \sin (4 x)) \sqrt{2}}{2}) + π}{16}$

Reorder factors in $\frac{4 \sin^{-1} (\frac{(\cos (4 x) - \sin (4 x)) \sqrt{2}}{2}) + π}{16}$ .

$g (\cos (4 x) - \sin (4 x)) = \frac{4 \sin^{-1} (\frac{\sqrt{2} (\cos (4 x) - \sin (4 x))}{2}) + π}{16}$

Since $g (f (x)) = x$ , $f^{- 1} (x) = \frac{4 \sin^{-1} (\frac{x \sqrt{2}}{2}) + π}{16}$ is the inverse of $f (x) = \cos (4 x) - \sin (4 x)$ .

$f^{- 1} (x) = \frac{4 \sin^{-1} (\frac{x \sqrt{2}}{2}) + π}{16}$

Find the Inverse cos(4x)-sin(4x)

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