# Simplify (tanx+secx-1)/(tanx-secx+1)?

Apr 29, 2016

#### Explanation:

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

Multiplying numerator and denominator by $\tan x + \sec x + 1$

= $\frac{\tan x + \sec x - 1}{\tan x - \sec x + 1} \times \frac{\tan x + \sec x + 1}{\tan x + \sec x + 1}$

= $\frac{{\tan}^{2} x + \tan x \sec x + \tan x + \tan x \sec x + {\sec}^{2} x + \sec x - \tan x - \sec x - 1}{{\tan}^{2} x + \tan x \sec x + \tan x - \tan x \sec x - {\sec}^{2} x - \sec x + \tan x + \sec x + 1}$

= $\frac{{\tan}^{2} x + 2 \tan x \sec x + {\sec}^{2} x - 1}{{\tan}^{2} x + 2 \tan x - {\sec}^{2} x + 1}$

As ${\sec}^{2} x = {\tan}^{2} x + 1$, above is equal to

= $\frac{{\tan}^{2} x + 2 \tan x \sec x + {\tan}^{2} x + 1 - 1}{{\tan}^{2} x + 2 \tan x - {\tan}^{2} x - 1 + 1}$

= $\frac{2 {\tan}^{2} x + 2 \tan x \sec x}{2 \tan x}$

= $\frac{2 \tan x \left(\tan x + \sec x\right)}{2 \tan x}$

= $\tan x + \sec x$