Trigonometry Examples

$\frac{1}{1 - \sec^{2} (x)}$

Step 1

Simplify with factoring out.

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Step 1.1

Reorder $1$ and $- \sec^{2} (x)$ .

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

Step 1.2

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

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

Step 1.3

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{1}{- \sec^{2} (x) - 1 \cdot - 1}$

Step 1.4

Factor $- 1$ out of $- \sec^{2} (x) - 1 \cdot - 1$ .

$\frac{1}{- (\sec^{2} (x) - 1)}$

Step 2

Apply pythagorean identity.

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.1.2

Move the negative in front of the fraction.

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

Step 3.2

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

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

Step 3.3

Rewrite $\frac{1^{2}}{\tan^{2} (x)}$ as ${(\frac{1}{\tan (x)})}^{2}$ .

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

Step 4

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

$- {(\frac{1}{\frac{\sin (x)}{\cos (x)}})}^{2}$

Step 5

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

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

Step 6

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

$- \cot^{2} (x)$