Question

Differential coefficient of `tan^-1(x/(1 + sqrt(1 - x^2)))` w.r.t. sin^-1 x is ______ 

Options

• `1/2`

• 1

• 2

• `3/2`

Answer 1

Differential coefficient of `tan^-1(x/(1 + sqrt(1 - x^2)))` w.r.t. sin^-1x is `underline(1/2)`.

Explanation:

Let y = `tan^-1(x/(1 + sqrt(1 - x^2)))` and z = sin^-1x

Put x = sin θ ⇒ θ = sin^-1x

∴ y = `tan^-1(sintheta/(1 + costheta))`

= `tan^-1(tantheta/2) = theta/2`

= `sin^-1x/2`

∴ `dy/dx = 1/(2sqrt(1 - x^2))` and `dz/dx = 1/sqrt(1 - x^2)`

∴ `dy/dz = (dy/dx)/(dz/dx) = 1/2`