Question

If 2tan^-1 (cos x) = tan^-1 (cosec² x), then x = ______.

Options

• `pi/3`

• `pi/2`

• `pi/6`

• π

Answer 1

`pi/3`

Explanation:

Given, 2tan^-1 (cos x) = tan^-1 (cosec² x)

`therefore tan^-1 ((2 cos x)/(1 - cos^2 x)) = tan^-1 (1/(sin^2 x))   .....[because 2 tan^-1 x = tan^-1 ("2x"/(1 - x^2))]`

`=> (2 cos x)/(1 - cos^2 x) = 1/(sin^2 x)`

`=> (2 cos x)/(sin^2 x) = 1/(sin^2 x)`

⇒ 2 cos x = 1

⇒ cos x = `1/2`

⇒ x = `pi/3`