Trigonometry Examples

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

Step 1

Set the denominator in $\frac{\csc (- x)}{1 + \tan^{2} (x)}$ equal to $0$ to find where the expression is undefined.

$1 + \tan^{2} (x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

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

Step 2.2

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

$\tan (x) = \pm \sqrt{- 1}$

Step 2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\tan (x) = \pm i$

Step 2.4

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

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Step 2.4.1

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

$\tan (x) = i$

Step 2.4.2

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

$\tan (x) = - i$

Step 2.4.3

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

$\tan (x) = i, - i$

Step 2.5

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

$\tan (x) = i$

$\tan (x) = - i$

Step 2.6

Solve for $x$ in $\tan (x) = i$ .

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Step 2.6.1

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

$x = \arctan (i)$

Step 2.6.2

The inverse tangent of $\arctan (i)$ is undefined.

Undefined

Step 2.7

Solve for $x$ in $\tan (x) = - i$ .

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Step 2.7.1

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

$x = \arctan (- i)$

Step 2.7.2

The inverse tangent of $\arctan (- i)$ is undefined.

Undefined

Step 2.8

List all of the solutions.

No solution

Step 3

Set the argument in $\csc (- x)$ equal to $π n$ to find where the expression is undefined.

$- x = π n$ , for any integer $n$

Step 4

Divide each term in $- x = π n$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- x = π n$ by $- 1$ .

$\frac{- x}{- 1} = \frac{π n}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{π n}{- 1}$

Step 4.2.2

Divide $x$ by $1$ .

$x = \frac{π n}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative one from the denominator of $\frac{π n}{- 1}$ .

$x = - 1 \cdot (π n)$

Step 4.3.2

Rewrite $- 1 \cdot (π n)$ as $- (π n)$ .

$x = - π n$

Step 5

Set the argument in $\tan (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

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

Step 6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

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

Step 7