Precalculus Examples

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

Step 1

Rewrite $\sec (x - 1)$ in terms of sines and cosines.

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

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x - 1)}$ .

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

Step 3

Multiply $\cos (x - 1)$ by $1$ .

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

Step 4

To write $- \cos (x - 1)$ as a fraction with a common denominator, multiply by $\frac{\sec (x + 1)}{\sec (x + 1)}$ .

$\frac{1}{\sec (x + 1)} - \cos (x - 1) \cdot \frac{\sec (x + 1)}{\sec (x + 1)}$

Step 5

Combine $- \cos (x - 1)$ and $\frac{\sec (x + 1)}{\sec (x + 1)}$ .

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

Step 6

Combine the numerators over the common denominator.

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