Question

Find `bb(dy/dx)` in the following:

y = `cos^(-1) ((1-x^2)/(1+x^2))`, 0 < x < 1

Answer 1

y = `cos^-1 ((1 - x^2)/(1 + x^2))`

Let, x = tan θ

⇒ θ = tan⁻¹ x

∴ y = `cos^-1 ((1 - tan^2 theta)/(1 + tan^2 theta))`

= cos⁻¹ (cos 2 θ)

= 2 θ

= 2 tan⁻¹ x

On differentiating with respect to x,

`dy/dx = 2 d/dx tan^-1 x`

`dy/dx = 2 xx 1/(1 + x^2)`

`dy/dx = 2/(1 + x^2)`