Question

Write the function in the simplest form:  `tan^(-1)  ((cos x - sin x)/(cos x + sin x)) `,` 0 < x < pi`

Answer 1

`tan^(-1)  (cos x - sin x)/(cos x + sin x)`

Dividing cos x inside

`= tan^(-1) [((cos x- sin x)/cos x)/((cos x + sinx)/cosx)]`

`= tan^(-1) [(cos x/cos x - sin x/cos x)/(cos x/cos x + sin x/cosx)]`

`= tan^(-1)  (1 - tan x)/(1 + tan x)`

`= tan^(-1)  [(1 - tan x)/(1+1. tan x)]`

`= tan^(-1) [(tan pi/4 - tan x)/(1 + tan pi/4 . tan x)]`     (As  tan `pi/4` = 1)

`= tan^(-1) tan (pi/4 - x)`

`= pi/4 - x`