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

To prove that

\[ \cot^{-1} \left( \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right) = \frac{x}{2}, \quad x \in \left(0, \frac{\pi}{4}\right), \]

we will simplify the left-hand side step by step.

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