Prove that `sin^(-1) {(sqrt(1 + x) + sqrt(1 - x))/(2)} = (pi)/(4) + (cos^(-1) x)/(2), 0 lt x lt 1`

Explanation

To prove the equation

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

we will follow these steps:

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