Question

Solve the following equation:

2 tan⁻¹ (cos x) = tan⁻¹ (2 cosec x)

Answer 1

`2 tan^(-1) (cos x) = tan^(-1) (2  "cosec"  x)`

⇒ `tan^(-1) ((2 cos x)/(1- cos^2 x)) = tan^(-1) (2  "cosec"  x)`   ...`[2 tan^(-1) x = tan^(-1)  (2x)/(1 - x^2)]`

⇒ `(2 cos x)/(1 - cos^2 x) = 2  "cosec"  x`

⇒`(2 cos x)/(sin^2 x) = 2/sin x`

⇒ cos x = sin x

⇒ tan x = 1

∴ `x = pi/4`