Question

If `y = sin^-1 ((sqrt(1 + x) + sqrt(1 - x))/2)`, then show that `dy/dx = (-1)/(2sqrt(1 - x^2)`.

Answer 1

To Prove: `dy/dx = (-1)/(2sqrt(1 - x^2)`

Given: `y = sin^-1 ((sqrt(1 + x) + sqrt(1 - x))/2)`

Put x = cos 2θ

`y = sin^-1 [(sqrt(1 + cos2θ) + sqrt(1 - cos2θ))/2]`

`y = sin^-1 [(sqrt(1 + 2 cos^2θ - 1) + sqrt(1 - (1 - 2 sin^2θ)))/2]`   ...[cos2θ = 2cos²θ – 1 = 1 – 2sin²θ]

`y = sin^-1 [(sqrt(2) cos θ + sqrt(2) sin θ)/2]`

`y = sin^-1 [1/sqrt(2) cos θ + 1/sqrt(2) sin θ]`

`y = sin^-1 [sin  π/4 cos θ + cos  π/4 sin θ]`   ...[Using sinA cosB + cosA sin B = sin(A + B)]

`y = sin^-1 [sin(π/4 + θ)]`

`y = π/4 + θ`

Put `θ = 1/2 cos^-1x`   ...[x = cos 2θ]

`y = π/4 + 1/2 cos^-1x`

Differentiate the above equation w.r.t x:

`dy/dx = 0 + 1/2 xx (-1)/sqrt(1 - x^2)`

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

Hence Proved.