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Question
Differentiate w.r.t. x the function:
`cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx) - sqrt(1-sinx))]`, ` 0 < x < pi/2`
Solution
Let, y = `cot^-1 [(sqrt(1 + sin x) + sqrt(1 - sin x))/(sqrt(1 + sin x) - sqrt(1 - sin x))]`
Now, `1 + sin x = sin^2 x/2 + cos^2 x/2 + 2 sin x/2 cos x/2`
`= (cos x/2 + sin x/2)`
`therefore sqrt(1 + sin x) = cos x/2 + sin x/2`
Similarly,
`sqrt(1 + sin x) = cos x/2 + sin x/2`
y = `cot^-1 [((cos x/2 + sin x/2) + (cos x/2 - sin x/2))/((cos x/2 + sin x/2) - (cos x/2 + sin x/2))]`
`= cot^-1 [(cos x/2 + sin x/2 + cos x/2 - sin x/2)/(cos x/2 + sin x/2 - cos x/2 + sin x/2)]`
`= cot^-1 [(2 cos x/2)/(2 sin x/2)]`
`= cot^-1 (cot x/2)`
y = `x/2`
On differentiating with respect to x,
`dy/dx = 1/2 * d/dx (x) = 1/2`
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