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Question
Prove that `cot^(-1) ((sqrt(1 + sin x) + sqrt(1 - sinx))/(sqrt(1 + sin x) - sqrt(1 - sinx))) = x/2, x ∈ (0, pi/4)`.
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Solution
Consider `(sqrt(1 + sinx) + sqrt(1 - sin x))/(sqrt(1 + sinx) - sqrt(1 - sinx))`
= `((sqrt(1 + sinx) + sqrt(1 - sinx))^2)/((sqrt(1 + sin x))^2 - (sqrt(1 - sin x))^2)` ...(By rationalizing)
= `((1 + sinx) + (1 - sinx) + 2sqrt((1 + sinx)(1 - sinx)))/(1 + sinx - 1 + sinx)`
= `(2(1 + sqrt(1 - sin^2x)))/(2sinx)`
= `(1 + cosx)/(sin x)`
= `(2 cos^2 x/2)/(2sin x/2 cos x/2)`
= `cot x/2`
∴ L.H.S = `cot^(-1) ((sqrt(1 + sin x) + sqrt(1 - sinx))/(sqrt(1 + sin x) - sqrt(1 - sinx)))`
= `cot^(-1) (cot x/2)`
= `x/2` = R.H.S.
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