Question

Simplify:

`tan^-1((cos 2x - sin 2x)/(cos 2x + sin 2x)), 0 < x < π/4`.

Answer 1

`tan^-1((cos 2x - sin 2x)/(cos 2x + sin 2x))`

Divide numerator and denominator by cos⁡ 2x:

`tan^-1 ((1 - tan 2x)/(1 + tan 2x))`

Using,

`tan(A - B) = (tan A - tan B)/(1 + tan A tan B)`

= `(1 - tan 2x)/(1 + tan 2x)`

= `tan (π/4 - 2x)`

Since, `0 < x < π/4`,

∴ `tan^-1((cos 2x - sin 2x)/(cos 2x + sin 2x)) = π/4 - 2x`