Question

If x ∈ R – {0}, then `tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2)))`

Options

• `1/2 cos^-1 (x^2)`

• `π/2 + 1/2 cos^-1 (x^2)`

• `π/4 + 1/2 cos^-1 (x^2)`

• None of these

Answer 1

`bb(π/4 + 1/2 cos^-1 (x^2))`

Explanation:

Let x² = cos 2θ `\implies 0 ≤ 2θ < π/2`  ...(∵ x² > 0)

The expression = `tan^-1((sqrt(2)cosθ + sqrt(2)sinθ)/(sqrt(2)cosθ - sqrt(2) sin θ))`

= `tan^-1((1 + tan θ)/(1 - tan θ))`

= `π/4 + θ`

= `π/4 + 1/2 cos^-1(x^2)`