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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 in (0, pi/4)`
Solution
Consider `((sqrt(1+sinx) + sqrt(1-sin x))/(sqrt(1+sinx) - sqrt(1-sinx))) = x/2` `x in (0, pi/4)`
`= ((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)))/(2sin x) `
`= (1+ cosx)/sin x = (2 cos^2 x/2)/(2sin x/2 cos x/2)`
`= cot^-1 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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