Trigonometry Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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Step 1.1.5.1

Cancel the common factor.

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

Step 1.1.5.2

Rewrite the expression.

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

Step 1.1.6

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

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

Step 1.2

Simplify with factoring out.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 2

Apply pythagorean identity.

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