Prove that `cot^(-1) ((sqrt(1 + sin x) + sqrt(1 - sin x))/(sqrt(1 + sin x) - sqrt(1 - sin x))) = (x)/(2), x in (0, (pi)/(4))`

Explanation

Consider `(sqrt(1 + sin x) + sqrt(1 - sin x))/(sqrt(1 + sin x) - sqrt(1 - sin x))`
`= ((sqrt(1 + sin x) + sqrt(1 - sin x))^(2))/((sqrt(1 + sin x)^(2)) - (sqrt(1 - sin x))^(2))` (by rationalizing)
`= ((1 + sin x) + (1 - sin x) + 2 sqrt((1 + sin x) (1 - sin x)))/(1 + sin x - 1 + sin x)`
`= (2(1 + sqrt(1 - sin^(2)x)))/(2 sin x) = (1 + cos x)/(sin x)`
`= (2 cos^(2) (x)/(2))/(2 sin (x)/(2) cos (x)/(2)) = cot. (x)/(2)`
`rArr cot^(-1) ((sqrt(1 + sin x)+ sqrt(1 - sin x))/(sqrt(1 + sin x) - sqrt(1 - sin x))) = (x)/(2)`