Simplify (csc(x)^2-1)/(cos(x)^2)

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

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 $\cos (x)$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Simplify the expression.

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

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

One to any power is one.

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

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 $\csc (x)$ .

$\csc^{2} (x)$