Question

Express `tan^-1 ((cos x - sin x)/(cos x + sin x))`, 0 < x < π in the simplest form.

Answer 1

Consider `((cos x - sin x)/(cos x + sin x))`

Dividing the numerator and denominator by cos x.

we get `(((cosx)/(cosx) - (sin x)/(cosx))/((cos x)/(cos x) + (sin x)/(cosx)))`

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

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

`[because tan  pi/4 = 1]`

`= tan (pi/4 - x) [because tan ("A - B") = (tan "A" - tan "B")/(1 + tan "A" tan "B")]`

`therefore tan^-1 [tan (pi/4 - x)]`

`= pi/4 - x`

which is the simplest form.