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