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)`.

Answer 1

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.