Question

Find `dy/dx`, if y = `sec^-1((1 + x^2)/(1 - x^2))`.

Answer 1

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

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

Put x = tan θ

`\implies` θ = tan^–1x

Put in (i)

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

= cos^–1(cos 2θ)

= 2θ

y = 2tan^–1xy

∴ `dy/dx = 2/(1 + x^2)`