Question

The simplest form of `tan^-1 [(sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x))]` is ______.

Options

• `π/4 - x/2`

• `π/4 + x/2`

• `π/4 - 1/2 cos^-1x`

• `π/4 + 1/2 cos^-1x`

Answer 1

The simplest form of `tan^-1 [(sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x))]` is `underlinebb(π/4 - 1/2 cos^-1x)`.

Explanation:

We have,

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

Put x = cos 2θ, so that `θ = 1/2 cos^-1x`

`tan^-1 ((sqrt(1 + cos 2θ) - sqrt(1 - cos 2θ))/(sqrt(1 + cos 2θ) + sqrt(1 - cos 2θ)))`

= `tan^-1 ((sqrt(2cos^2θ) - sqrt(2sin^2θ))/(sqrt(2cos^2θ) + sqrt(2sin^2θ)))`

= `tan^-1 ((cos θ - sin θ)/(cos θ + sin θ))`

= `tan^-1 ((1 - tan θ)/(1 + tan θ))`

= tan^–1(1) – tan^–1(tan θ)  ...`[∵ tan^-1 ((x - y)/(1 + xy)) = tan^-1x - tan^-1y]`

= `tan^-1(tan  π/4) - θ`

= `π/4 - 1/2 cos^-1x`