Trigonometry Examples

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

Step 1

Apply pythagorean identity.

$\frac{\cot^{2} (x)}{\cos^{2} (x)}$

Step 2

Rewrite $\frac{\cot^{2} (x)}{\cos^{2} (x)}$ as ${(\frac{\cot (x)}{\cos (x)})}^{2}$ .

${(\frac{\cot (x)}{\cos (x)})}^{2}$

Step 3

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

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

Step 4

Rewrite $\frac{\frac{\cos (x)}{\sin (x)}}{\cos (x)}$ as a product.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Simplify the expression.

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Step 6.1

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 6.2

One to any power is one.

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

$\csc^{2} (x)$