If `piltxlt2pi,` prove that `(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))=cot(x/2+pi/4)dot`

Explanation

To prove the equation

\[ \frac{\sqrt{1+\cos x} + \sqrt{1-\cos x}}{\sqrt{1+\cos x} - \sqrt{1-\cos x}} = \cot\left(\frac{x}{2} + \frac{\pi}{4}\right) \]

given that \(x\) lies between \(\pi\) and \(2\pi\), we will start with the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS).

...