Question

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

Answer 1

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

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

= `sqrt((2cos^2(θ/2))/(2sin^2(θ/2)`

= `sqrt(cot^2(θ/2)`

= `cot(θ/2)`

∴ `cot^-1sqrt((1 + x)/(1 - x))`

= `cot^-1[cot(θ/2)]`

= `θ/(2)`

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

∴ y = `sin^2(1/2 cos^-1x)`

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

= `2sin(1/2cos^-1x)."d"/"dx"sin(1/2cos^-1x)`

= `2sin(1/2cos^-1x).cos(1/2cos^-1x)."d"/"dx"(1/2cos^-1x)`

= `sin[2(1/2cos^-1x)] xx (1)/(2)."d"/"dx"(cos^-1x)`

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

= `sin(sin^-1sqrt(1 - x^2)) xx (-1)/(2sqrt(1 - x^2))   ...[∵ cos^-1x = sin^-1 sqrt(1 - x^2)]`

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

= `-(1)/(2)`.