Question

Differentiate `tan^(-1)  sqrt(1 - x^2)/x  w.r.t. cos^(-1)(2xsqrt(1 - x ^2)), x ∈ (1/(sqrt2), 1)`.

Answer 1

Let u = `tan^(-1)  sqrt(1 - x^2)/x`

Put x = cos θ, θ = cos⁻¹ x

⇒ u = `tan^(-1)  sqrt(1 - cos^2 θ)/cos θ`

⇒ u = `tan^(-1) ((sin θ)/cos θ)`

⇒ u = tan⁻¹ (tan θ)

⇒ u = θ

∴ u =  cos⁻¹ x

`(du)/(dx) = (-1)/(sqrt(1 - x^2))    ...(i)`

Again let, v = `cos^(-1)(2xsqrt(1 - x ^2))`

Put x = sin θ, θ = sin⁻¹ x

⇒ v = `cos^(-1)(2 sin θ sqrt(1 - sin^2θ))`

⇒ v = cos⁻¹ (2 sin θ cos θ)

= cos⁻¹ (sin 2θ)

⇒ v = `cos^(-1)[cos(pi/2 - 2θ)]`

⇒ v = `pi/2 - 2θ`

∴ v = `pi/2 - 2 sin^(-1) x`

`(dv)/(dx) = (-2)/sqrt(1 - x^2)    ...(ii)`

Dividing equation (i) by (ii), we get

`(du)/(dv) = (du//dx)/(dv//dx)`

= `((-1)/sqrt(1 - x^2))/((-2)/sqrt(1 - x^2))`

= `1/2`