Prove that: `tan^(-1)[(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2cos^(-1)x ,\ -1/(sqrt(2))\ lt=x\ lt=1`

Explanation

To prove the identity

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

we will start with the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS).

...