Question

Prove that:

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

Answer 1

Let tan^-1 x = θ

x = tan θ

sin 2θ = `(2 tan θ)/(1 + tan^2 θ) = "2x"/(1 + x^2)`

2θ = `sin^-1  ("2x"/(1 + x^2))`

∴ 2 tan^-1 x = `sin^-1  ("2x"/(1 + x^2))` = RHS