Question

Differentiate the following w.r.t. x : `sin[2tan^-1(sqrt((1 - x)/(1 + x)))]`

Answer 1

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

Put x = cosθ. Thenθ = cos^–1x and
`sqrt((1 - x)/(1 + x)) = sqrt((1 - cosθ)/(1 + cosθ)`

= `sqrt((2sin^2(θ/2))/(2cos^2(θ/2)`

= `sqrt(tan^2(θ/2)`

= `tan(θ/2)`

∴ `tan^-1(sqrt((1 - x)/(1 + x)))`

= `tan^-1[tan(θ/2)]`

= `θ/(2)`

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

∴ y = `sin[2 xx 1/2 cos^-1 x]`
= sin (cos^–1x)
∴ `"dy"/"dx" = "d"/"dx"[sin(cos-1x)]`

= `cos(cos^-1x)."d"/"dx"(cos^-1x)`

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

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