Question

If tan⁻¹ `((sqrt(1 + x^2) − sqrt(1 − x^2))/(sqrt(1 + x^2) + sqrt(1 − x^2)))` = α, prove that sin 2 α = x².

Answer 1

Let x² = cos 2 θ

tan⁻¹ `((sqrt(1 + cos 2 θ) − sqrt(1 − cos 2 θ))/(sqrt(1 + cos 2 θ) + sqrt(1 − cos 2 θ)))` = α

tan⁻¹ `((cos θ − sin θ)/(cos θ − sin θ))`

= tan⁻¹ `((1 − tan θ)/(1 + tan θ))`

= tan⁻¹ `(tan(π/4 − θ))`

`π/4` − θ = α

`π/2 − 2θ = 2α`

sin`(π/2 − 2θ) = sin 2 α`

cos 2 θ = sin 2 α

x² = sin 2 α

Hence Proved