Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\frac{\frac{1}{\cos (x)}}{\sin (x)}$ as a product.

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

Step 5

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\sin (x)}$ .

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

Step 6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.1

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

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

Step 6.2

Apply the product rule to $\cos (x) \sin (x)$ .

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

Step 7

One to any power is one.

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

Step 8

Multiply by $1$ .

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

Step 9

Separate fractions.

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

Step 10

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

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

Step 11

Write the expression using exponents.

Tap for more steps...

Step 11.1

Multiply $\sin^{2} (x)$ by $1$ .

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

Step 11.2

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

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

Step 12

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

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

Step 13

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

$\csc^{2} (x) \sec^{2} (x)$