Prove that (cos x + sin x) / (cos x - sin x) = tan [x+(pi/4)]

The identity to be proven is (cos x + sin x) / (cos x -
sin x) = tan [x+(pi/4)]

Start from the right hand
side

tan [x+(pi/4)]

expand tan
(x + pi/4)

=> (tan x + tan pi/4) / (1 - tan x* tan
pi/4)

use tan pi/4 =
1

=> (tan x + 1) / (1 - tan
x)

substitute tan x = sin x/ cos
x

= [(sin x / cos x) + 1] / [1 - (sin x / cos
x)]

= [(sin x / cos x) + (cos x / cos x)] / [(cos x / cos
x) - (sin x / cos x)]

= [(sin x + cos x) / cos x] / [(cos x
- sin x) / cos x)]

= [(sin x + cos x) / cos x] * [cos x /
(cos x - sin x)]

= (sin x + cos x) / (cos x - sin
x)

This is the left hand
side.

This proves (cos x + sin x) / (cos x -
sin x) = tan [x+(pi/4)]