Solve for x tan(2x)tan(x)=1

$\tan (2 x) \tan (x) = 1$

Divide each term by $\tan (x)$ and simplify.

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Divide each term in $\tan (2 x) \tan (x) = 1$ by $\tan (x)$ .

$\frac{\tan (2 x) \tan (x)}{\tan (x)} = \frac{1}{\tan (x)}$

Cancel the common factor of $\tan (x)$ .

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Cancel the common factor.

$\frac{\tan (2 x) \tan (x)}{\tan (x)} = \frac{1}{\tan (x)}$

Divide $\tan (2 x)$ by $1$ .

$\tan (2 x) = \frac{1}{\tan (x)}$

Simplify $\frac{1}{\tan (x)}$ .

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Rewrite $\tan (x)$ in terms of sines and cosines.

$\tan (2 x) = \frac{1}{\frac{\sin (x)}{\cos (x)}}$

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$\tan (2 x) = \frac{\cos (x)}{\sin (x)}$

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\tan (2 x) = \cot (x)$

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

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Subtract $\cot (x)$ from both sides of the equation.

$\tan (2 x) - \cot (x) = 0$

Simplify each term.

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Apply the tangent double–angle identity.

$\frac{2 \tan (x)}{1 - \tan^{2} (x)} - \cot (x) = 0$

Simplify the denominator.

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Rewrite $1$ as $1^{2}$ .

$\frac{2 \tan (x)}{1^{2} - \tan^{2} (x)} - \cot (x) = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \tan (x)$ .

$\frac{2 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))} - \cot (x) = 0$

Substitute $u$ for $\tan (x)$ .

$\frac{2 (u)}{(1 + u) (1 - (u))} - \cot (x) = 0$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(1 + u) (1 - u), 1, 1$

The LCM is the smallest number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $1 + u$ is $1 + u$ itself.

$(1 + W) = 1 + u$

$(1 + W)$ occurs $1$ time.

The factor for $1 - u$ is $1 - u$ itself.

$(1 - W) = 1 - u$

$(1 - W)$ occurs $1$ time.

The LCM of $1 + u, 1 - u$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(1 + u) (1 - u)$

Multiply each term by $(1 + u) (1 - u)$ and simplify.

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Multiply each term in $\frac{2 u}{(1 + u) (1 - u)} - \cot (x) = 0$ by $(1 + u) (1 - u)$ in order to remove all the denominators from the equation.

$\frac{2 u}{(1 + u) (1 - u)} \cdot ((1 + u) (1 - u)) - \cot (x) \cdot ((1 + u) (1 - u)) = 0 \cdot ((1 + u) (1 - u))$

Simplify each term.

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Cancel the common factor of $(1 + u) (1 - u)$ .

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Cancel the common factor.

$\frac{2 u}{(1 + u) (1 - u)} \cdot ((1 + u) (1 - u)) - \cot (x) \cdot ((1 + u) (1 - u)) = 0 \cdot ((1 + u) (1 - u))$

Rewrite the expression.

$2 u - \cot (x) \cdot ((1 + u) (1 - u)) = 0 \cdot ((1 + u) (1 - u))$

Expand $(1 + u) (1 - u)$ using the FOIL Method.

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Apply the distributive property.

$2 u - \cot (x) \cdot (1 (1 - u) + u (1 - u)) = 0 \cdot ((1 + u) (1 - u))$

Apply the distributive property.

$2 u - \cot (x) \cdot (1 \cdot 1 + 1 (- u) + u (1 - u)) = 0 \cdot ((1 + u) (1 - u))$

Apply the distributive property.

$2 u - \cot (x) \cdot (1 \cdot 1 + 1 (- u) + u \cdot 1 + u (- u)) = 0 \cdot ((1 + u) (1 - u))$

Simplify and combine like terms.

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Simplify each term.

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Multiply $1$ by $1$ .

$2 u - \cot (x) \cdot (1 + 1 (- u) + u \cdot 1 + u (- u)) = 0 \cdot ((1 + u) (1 - u))$

Multiply $- u$ by $1$ .

$2 u - \cot (x) \cdot (1 - u + u \cdot 1 + u (- u)) = 0 \cdot ((1 + u) (1 - u))$

Multiply $u$ by $1$ .

$2 u - \cot (x) \cdot (1 - u + u + u (- u)) = 0 \cdot ((1 + u) (1 - u))$

Rewrite using the commutative property of multiplication.

$2 u - \cot (x) \cdot (1 - u + u - u \cdot u) = 0 \cdot ((1 + u) (1 - u))$

Multiply $u$ by $u$ by adding the exponents.

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Move $u$ .

$2 u - \cot (x) \cdot (1 - u + u - (u \cdot u)) = 0 \cdot ((1 + u) (1 - u))$

Multiply $u$ by $u$ .

$2 u - \cot (x) \cdot (1 - u + u - u^{2}) = 0 \cdot ((1 + u) (1 - u))$

Add $- u$ and $u$ .

$2 u - \cot (x) \cdot (1 + 0 - u^{2}) = 0 \cdot ((1 + u) (1 - u))$

Add $1$ and $0$ .

$2 u - \cot (x) \cdot (1 - u^{2}) = 0 \cdot ((1 + u) (1 - u))$

Apply the distributive property.

$2 u - \cot (x) \cdot 1 - \cot (x) (- u^{2}) = 0 \cdot ((1 + u) (1 - u))$

Multiply $- 1$ by $1$ .

$2 u - \cot (x) - \cot (x) (- u^{2}) = 0 \cdot ((1 + u) (1 - u))$

Rewrite using the commutative property of multiplication.

$2 u - \cot (x) - 1 \cdot (- 1 (\cot (x) u^{2})) = 0 \cdot ((1 + u) (1 - u))$

Simplify each term.

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Multiply $- 1$ by $- 1$ .

$2 u - \cot (x) + 1 (\cot (x) u^{2}) = 0 \cdot ((1 + u) (1 - u))$

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

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot ((1 + u) (1 - u))$

Simplify $0 \cdot ((1 + u) (1 - u))$ .

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Expand $(1 + u) (1 - u)$ using the FOIL Method.

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Apply the distributive property.

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 (1 - u) + u (1 - u))$

Apply the distributive property.

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 \cdot 1 + 1 (- u) + u (1 - u))$

Apply the distributive property.

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 \cdot 1 + 1 (- u) + u \cdot 1 + u (- u))$

Simplify and combine like terms.

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Simplify each term.

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Multiply $1$ by $1$ .

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 + 1 (- u) + u \cdot 1 + u (- u))$

Multiply $- u$ by $1$ .

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 - u + u \cdot 1 + u (- u))$

Multiply $u$ by $1$ .

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 - u + u + u (- u))$

Rewrite using the commutative property of multiplication.

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 - u + u - u \cdot u)$

Multiply $u$ by $u$ by adding the exponents.

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Move $u$ .

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 - u + u - (u \cdot u))$

Multiply $u$ by $u$ .

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 - u + u - u^{2})$

Add $- u$ and $u$ .

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 + 0 - u^{2})$

Add $1$ and $0$ .

$2 u - \cot (x) + \cot (x) u^{2} = 0 \cdot (1 - u^{2})$

Multiply $0$ by $1 - u^{2}$ .

$2 u - \cot (x) + \cot (x) u^{2} = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = \cot (x)$ , $b = 2$ , and $c = - \cot (x)$ into the quadratic formula and solve for $u$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (\cot (x) \cdot (- \cot (x)))}}{2 \cot (x)}$

Simplify.

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Simplify the numerator.

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

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot (\cot (x) \cdot (- \cot (x)))}}{2 \cot (x)}$

Multiply $\cot (x)$ by $\cot (x)$ by adding the exponents.

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Move $\cot (x)$ .

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

Multiply $\cot (x)$ by $\cot (x)$ .

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

Move $- 1$ to the left of $\cot^{2} (x)$ .

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

Rewrite $- 1 \cot^{2} (x)$ as $- \cot^{2} (x)$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot (- \cot^{2} (x))}}{2 \cot (x)}$

Multiply $- 1$ by $- 4$ .

$u = \frac{- 2 \pm \sqrt{4 + 4 \cot^{2} (x)}}{2 \cot (x)}$

Factor $4$ out of $4 + 4 \cot^{2} (x)$ .

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Factor $4$ out of $4$ .

$u = \frac{- 2 \pm \sqrt{4 \cdot 1 + 4 \cot^{2} (x)}}{2 \cot (x)}$

Factor $4$ out of $4 \cdot 1 + 4 \cot^{2} (x)$ .

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

Rewrite $4 (1 + \cot^{2} (x))$ as $2^{2} (1^{2} + \cot^{2} (x))$ .

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Rewrite $4$ as $2^{2}$ .

$u = \frac{- 2 \pm \sqrt{2^{2} (1 + \cot^{2} (x))}}{2 \cot (x)}$

Rewrite $1$ as $1^{2}$ .

$u = \frac{- 2 \pm \sqrt{2^{2} (1^{2} + \cot^{2} (x))}}{2 \cot (x)}$

Pull terms out from under the radical.

$u = \frac{- 2 \pm 2 \sqrt{1^{2} + \cot^{2} (x)}}{2 \cot (x)}$

One to any power is one.

$u = \frac{- 2 \pm 2 \sqrt{1 + \cot^{2} (x)}}{2 \cot (x)}$

Simplify $\frac{- 2 \pm 2 \sqrt{1 + \cot^{2} (x)}}{2 \cot (x)}$ .

$u = \frac{- 1 \pm \sqrt{1 + \cot^{2} (x)}}{\cot (x)}$

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

$u = \frac{- 11 \pm \sqrt{1 + \cot^{2} (x)}}{\cot (x)}$

Multiply $- 1$ by $- 1$ .

$u = \frac{1 - 1 \pm \sqrt{1 + \cot^{2} (x)}}{\cot (x)}$

Multiply $- 1 \pm \sqrt{1 + \cot^{2} (x)}$ by $1$ .

$u = \frac{- 1 \pm \sqrt{1 + \cot^{2} (x)}}{\cot (x)}$

The final answer is the combination of both solutions.

$u = - \frac{1 - \sqrt{1 + \cot^{2} (x)}}{\cot (x)}$

$u = - \frac{1 + \sqrt{\cot^{2} (x) + 1}}{\cot (x)}$

Substitute $\tan (x)$ for $u$ .

$\tan (x) = - \frac{1 - \sqrt{1 + \cot^{2} (x)}}{\cot (x)}, - \frac{1 + \sqrt{\cot^{2} (x) + 1}}{\cot (x)}$

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

$\tan (x) = - \frac{1 - \sqrt{1 + \cot^{2} (x)}}{\cot (x)}$

$\tan (x) = - \frac{1 + \sqrt{\cot^{2} (x) + 1}}{\cot (x)}$

Set up the equation to solve for $x$ .

$\tan (x) = - \frac{1 - \sqrt{1 + \cot^{2} (x)}}{\cot (x)}$

Solve the equation for $x$ .

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Simplify the numerator.

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Apply pythagorean identity.

$\tan (x) = - \frac{1 - \sqrt{\csc^{2} (x)}}{\cot (x)}$

Pull terms out from under the radical, assuming positive real numbers.

$\tan (x) = - \frac{1 - \csc (x)}{\cot (x)}$

Rewrite the equation as $- \frac{1 - \csc (x)}{\cot (x)} = \tan (x)$ .

$- \frac{1 - \csc (x)}{\cot (x)} = \tan (x)$

Solve for $\cot (x)$ .

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Multiply each term by $\cot (x)$ and simplify.

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Multiply each term in $- \frac{1 - \csc (x)}{\cot (x)} = \tan (x)$ by $\cot (x)$ .

$(- \frac{1 - \csc (x)}{\cot (x)}) \cdot \cot (x) = \tan (x) \cdot \cot (x)$

Simplify the left side of the equation by cancelling the common factors.

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Cancel the common factor of $\cot (x)$ .

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Move the leading negative in $- \frac{1 - \csc (x)}{\cot (x)}$ into the numerator.

$\frac{- (1 - \csc (x))}{\cot (x)} \cdot \cot (x) = \tan (x) \cdot \cot (x)$

Cancel the common factor.

$\frac{- (1 - \csc (x))}{\cot (x)} \cdot \cot (x) = \tan (x) \cdot \cot (x)$

Rewrite the expression.

$- (1 - \csc (x)) = \tan (x) \cdot \cot (x)$

Apply the distributive property.

$- 1 \cdot 1 + \csc (x) = \tan (x) \cdot \cot (x)$

Multiply $- 1$ by $1$ .

$- 1 + \csc (x) = \tan (x) \cdot \cot (x)$

$- 1 + \csc (x) = \tan (x) \cot (x)$

Rewrite the equation as $\tan (x) \cot (x) = - 1 + \csc (x)$ .

$\tan (x) \cot (x) = - 1 + \csc (x)$

Divide each term by $\tan (x)$ and simplify.

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Divide each term in $\tan (x) \cot (x) = - 1 + \csc (x)$ by $\tan (x)$ .

$\frac{\tan (x) \cot (x)}{\tan (x)} = \frac{- 1}{\tan (x)} + \frac{\csc (x)}{\tan (x)}$

Cancel the common factor of $\tan (x)$ .

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Cancel the common factor.

$\frac{\tan (x) \cot (x)}{\tan (x)} = \frac{- 1}{\tan (x)} + \frac{\csc (x)}{\tan (x)}$

Divide $\cot (x)$ by $1$ .

$\cot (x) = \frac{- 1}{\tan (x)} + \frac{\csc (x)}{\tan (x)}$

Move the negative in front of the fraction.

$\cot (x) = - \frac{1}{\tan (x)} + \frac{\csc (x)}{\tan (x)}$

Simplify each term.

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Rewrite $\tan (x)$ in terms of sines and cosines.

$\cot (x) = - \frac{1}{\frac{\sin (x)}{\cos (x)}} + \frac{\csc (x)}{\tan (x)}$

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$\cot (x) = - \frac{\cos (x)}{\sin (x)} + \frac{\csc (x)}{\tan (x)}$

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\cot (x) = - \cot (x) + \frac{\csc (x)}{\tan (x)}$

Rewrite $\tan (x)$ in terms of sines and cosines.

$\cot (x) = - \cot (x) + \frac{\csc (x)}{\frac{\sin (x)}{\cos (x)}}$

Rewrite $\csc (x)$ in terms of sines and cosines.

$\cot (x) = - \cot (x) + \frac{\frac{1}{\sin (x)}}{\frac{\sin (x)}{\cos (x)}}$

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$\cot (x) = - \cot (x) + \frac{1}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}$

Simplify.

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Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\cot (x) = - \cot (x) + \csc (x) (\frac{\cos (x)}{\sin (x)})$

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\cot (x) = - \cot (x) + \csc (x) \cot (x)$

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

$- \cot (x) + \csc (x) \cot (x) = \cot (x)$

Factor $\cot (x)$ out of $\csc (x) \cot (x) - 2 \cot (x)$ .

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Factor $\cot (x)$ out of $\csc (x) \cot (x)$ .

$\cot (x) \csc (x) - 2 \cot (x) = 0$

Factor $\cot (x)$ out of $- 2 \cot (x)$ .

$\cot (x) \csc (x) + \cot (x) \cdot - 2 = 0$

Factor $\cot (x)$ out of $\cot (x) \csc (x) + \cot (x) \cdot - 2$ .

$\cot (x) (\csc (x) - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cot (x) = 0$

$\csc (x) - 2 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$\cot (x) = 0$

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

$x = arccot (0)$

The exact value of $arccot (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{2}$

Simplify $π + \frac{π}{2}$ .

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To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{π}{1} \cdot \frac{2}{2} + \frac{π}{2}$

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

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Combine.

$x = \frac{π \cdot 2}{1 \cdot 2} + \frac{π}{2}$

Multiply $2$ by $1$ .

$x = \frac{π \cdot 2}{2} + \frac{π}{2}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Simplify the numerator.

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Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Find the period.

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

$\frac{π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

The period of the $\cot (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{π}{2} + π n, \frac{3 π}{2} + π n$ , for any integer $n$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$\csc (x) - 2 = 0$

Add $2$ to both sides of the equation.

$\csc (x) = 2$

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

$x = arccsc (2)$

The exact value of $arccsc (2)$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

The cosecant 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.

$x = π - \frac{π}{6}$

Simplify $π - \frac{π}{6}$ .

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To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = \frac{π}{1} \cdot \frac{6}{6} - \frac{π}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{π \cdot 6}{1 \cdot 6} - \frac{π}{6}$

Multiply $6$ by $1$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Simplify the numerator.

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Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Find the period.

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

The period of the $\csc (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

The final solution is all the values that make $\cot (x) (\csc (x) - 2) = 0$ true.

$x = \frac{π}{2} + π n, \frac{3 π}{2} + π n, \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Set up the equation to solve for $x$ .

$\tan (x) = - \frac{1 + \sqrt{\cot^{2} (x) + 1}}{\cot (x)}$

Solve the equation for $x$ .

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Simplify the numerator.

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Rearrange terms.

$\tan (x) = - \frac{1 + \sqrt{1 + \cot^{2} (x)}}{\cot (x)}$

Apply pythagorean identity.

$\tan (x) = - \frac{1 + \sqrt{\csc^{2} (x)}}{\cot (x)}$

Pull terms out from under the radical, assuming positive real numbers.

$\tan (x) = - \frac{1 + \csc (x)}{\cot (x)}$

Rewrite the equation as $- \frac{1 + \csc (x)}{\cot (x)} = \tan (x)$ .

$- \frac{1 + \csc (x)}{\cot (x)} = \tan (x)$

Solve for $\cot (x)$ .

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Multiply each term by $\cot (x)$ and simplify.

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Multiply each term in $- \frac{1 + \csc (x)}{\cot (x)} = \tan (x)$ by $\cot (x)$ .

$(- \frac{1 + \csc (x)}{\cot (x)}) \cdot \cot (x) = \tan (x) \cdot \cot (x)$

Simplify the left side of the equation by cancelling the common factors.

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Cancel the common factor of $\cot (x)$ .

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Move the leading negative in $- \frac{1 + \csc (x)}{\cot (x)}$ into the numerator.

$\frac{- (1 + \csc (x))}{\cot (x)} \cdot \cot (x) = \tan (x) \cdot \cot (x)$

Cancel the common factor.

$\frac{- (1 + \csc (x))}{\cot (x)} \cdot \cot (x) = \tan (x) \cdot \cot (x)$

Rewrite the expression.

$- (1 + \csc (x)) = \tan (x) \cdot \cot (x)$

Apply the distributive property.

$- 1 \cdot 1 - \csc (x) = \tan (x) \cdot \cot (x)$

Multiply $- 1$ by $1$ .

$- 1 - \csc (x) = \tan (x) \cdot \cot (x)$

$- 1 - \csc (x) = \tan (x) \cot (x)$

Rewrite the equation as $\tan (x) \cot (x) = - 1 - \csc (x)$ .

$\tan (x) \cot (x) = - 1 - \csc (x)$

Divide each term by $\tan (x)$ and simplify.

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Divide each term in $\tan (x) \cot (x) = - 1 - \csc (x)$ by $\tan (x)$ .

$\frac{\tan (x) \cot (x)}{\tan (x)} = \frac{- 1}{\tan (x)} + \frac{- \csc (x)}{\tan (x)}$

Cancel the common factor of $\tan (x)$ .

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Cancel the common factor.

$\frac{\tan (x) \cot (x)}{\tan (x)} = \frac{- 1}{\tan (x)} + \frac{- \csc (x)}{\tan (x)}$

Divide $\cot (x)$ by $1$ .

$\cot (x) = \frac{- 1}{\tan (x)} + \frac{- \csc (x)}{\tan (x)}$

Simplify each term.

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Move the negative in front of the fraction.

$\cot (x) = - \frac{1}{\tan (x)} + \frac{- \csc (x)}{\tan (x)}$

Move the negative in front of the fraction.

$\cot (x) = - \frac{1}{\tan (x)} - \frac{\csc (x)}{\tan (x)}$

Simplify each term.

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Rewrite $\tan (x)$ in terms of sines and cosines.

$\cot (x) = - \frac{1}{\frac{\sin (x)}{\cos (x)}} - \frac{\csc (x)}{\tan (x)}$

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$\cot (x) = - \frac{\cos (x)}{\sin (x)} - \frac{\csc (x)}{\tan (x)}$

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\cot (x) = - \cot (x) - \frac{\csc (x)}{\tan (x)}$

Rewrite $\tan (x)$ in terms of sines and cosines.

$\cot (x) = - \cot (x) - \frac{\csc (x)}{\frac{\sin (x)}{\cos (x)}}$

Rewrite $\csc (x)$ in terms of sines and cosines.

$\cot (x) = - \cot (x) - \frac{\frac{1}{\sin (x)}}{\frac{\sin (x)}{\cos (x)}}$

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$\cot (x) = - \cot (x) - (\frac{1}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)})$

Simplify.

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Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\cot (x) = - \cot (x) - (\csc (x) (\frac{\cos (x)}{\sin (x)}))$

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\cot (x) = - \cot (x) - (\csc (x) \cot (x))$

$\cot (x) = - \cot (x) - \csc (x) \cot (x)$

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

$- \cot (x) - \csc (x) \cot (x) = \cot (x)$

Factor $\cot (x)$ out of $- \csc (x) \cot (x) - 2 \cot (x)$ .

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Factor $\cot (x)$ out of $- \csc (x) \cot (x)$ .

$\cot (x) (- \csc (x)) - 2 \cot (x) = 0$

Factor $\cot (x)$ out of $- 2 \cot (x)$ .

$\cot (x) (- \csc (x)) + \cot (x) \cdot - 2 = 0$

Factor $\cot (x)$ out of $\cot (x) (- \csc (x)) + \cot (x) \cdot - 2$ .

$\cot (x) (- \csc (x) - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cot (x) = 0$

$- \csc (x) - 2 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$\cot (x) = 0$

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

$x = arccot (0)$

The exact value of $arccot (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{2}$

Simplify $π + \frac{π}{2}$ .

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To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{π}{1} \cdot \frac{2}{2} + \frac{π}{2}$

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

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Combine.

$x = \frac{π \cdot 2}{1 \cdot 2} + \frac{π}{2}$

Multiply $2$ by $1$ .

$x = \frac{π \cdot 2}{2} + \frac{π}{2}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Simplify the numerator.

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Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Find the period.

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

$\frac{π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

The period of the $\cot (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{π}{2} + π n, \frac{3 π}{2} + π n$ , for any integer $n$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$- \csc (x) - 2 = 0$

Add $2$ to both sides of the equation.

$- \csc (x) = 2$

Multiply each term in $- \csc (x) = 2$ by $- 1$

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Multiply each term in $- \csc (x) = 2$ by $- 1$ .

$(- \csc (x)) \cdot - 1 = 2 \cdot - 1$

Multiply $- \csc (x) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 \csc (x) = 2 \cdot - 1$

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

$\csc (x) = 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$\csc (x) = - 2$

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

$x = arccsc (- 2)$

The exact value of $arccsc (- 2)$ is $- \frac{π}{6}$ .

$x = - \frac{π}{6}$

The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{6} + π$

Simplify the expression to find the second solution.

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Simplify $2 π + \frac{π}{6} + π$ .

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = \frac{2 π}{1} \cdot \frac{6}{6} + \frac{π}{6} + π$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{2 π \cdot 6}{1 \cdot 6} + \frac{π}{6} + π$

Multiply $6$ by $1$ .

$x = \frac{2 π \cdot 6}{6} + \frac{π}{6} + π$

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 + π}{6} + π$

Simplify the numerator.

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Multiply $6$ by $2$ .

$x = \frac{12 π + π}{6} + π$

Add $12 π$ and $π$ .

$x = \frac{13 π}{6} + π$

To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = \frac{13 π}{6} + \frac{π}{1} \cdot \frac{6}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{13 π}{6} + \frac{π \cdot 6}{1 \cdot 6}$

Multiply $6$ by $1$ .

$x = \frac{13 π}{6} + \frac{π \cdot 6}{6}$

Combine the numerators over the common denominator.

$x = \frac{13 π + π \cdot 6}{6}$

Simplify the numerator.

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Move $6$ to the left of $π$ .

$x = \frac{13 π + 6 \cdot π}{6}$

Add $13 π$ and $6 π$ .

$x = \frac{19 π}{6}$

Subtract $2 π$ from $\frac{19 π}{6}$ .

$x = \frac{19 π}{6} - 2 π$

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $\frac{19 π}{6}$ .

$x = \frac{7 π}{6}$

Find the period.

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Add $2 π$ to every negative angle to get positive angles.

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Add $2 π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{2 π}{1} \cdot \frac{6}{6} - \frac{π}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 π \cdot 6}{1 \cdot 6} - \frac{π}{6}$

Multiply $6$ by $1$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Simplify the numerator.

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Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

List the new angles.

$x = \frac{11 π}{6}$

The period of the $\csc (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

The final solution is all the values that make $\cot (x) (- \csc (x) - 2) = 0$ true.

$x = \frac{π}{2} + π n, \frac{3 π}{2} + π n, \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

List all of the results found in the previous steps.

$x = \frac{π}{6} + 2 π n, \frac{π}{2} + π n, \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n, \frac{3 π}{2} + π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

The complete solution is the set of all solutions.

$x = \frac{π}{2} + π n, \frac{3 π}{2} + π n, \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{π}{2} + π n, \frac{3 π}{2} + π n, \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Consolidate the answers.

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Consolidate $\frac{π}{2} + π n$ and $\frac{3 π}{2} + π n$ to $\frac{π}{2} + π n$ .

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{π}{2} + π n, \frac{3 π}{2} + π n, \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Consolidate $\frac{π}{6} + 2 π n$ and $\frac{7 π}{6} + 2 π n$ to $\frac{π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + 2 π n, \frac{π}{2} + π n, \frac{3 π}{2} + π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Consolidate $\frac{5 π}{6} + 2 π n$ and $\frac{11 π}{6} + 2 π n$ to $\frac{5 π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + π n, \frac{π}{2} + π n, \frac{3 π}{2} + π n$ , for any integer $n$

Consolidate $\frac{π}{2} + π n$ and $\frac{3 π}{2} + π n$ to $\frac{π}{2} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + π n, \frac{π}{2} + π n$ , for any integer $n$

Consolidate the answers.

$x = \frac{π}{6} + \frac{π n}{3}$ , for any integer $n$

Verify each of the solutions by substituting them into $\tan (2 x) \tan (x) = 1$ and solving.

$x = \frac{π}{6} + π n, \frac{5 π}{6} + π n$ , for any integer $n$

Solve for x tan(2x)tan(x)=1

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