# How do you simplify cos x /( sec x -1) + cos x / (sec x +1)?

Jul 8, 2016

$2 {\cot}^{2} x$

#### Explanation:

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

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

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

$= \frac{\cos x \cdot 2 \sec x}{{\sec}^{2} x - 1}$

$= \frac{2}{\tan} ^ 2 x = 2 {\cot}^{2} x$