Question

Write ${\tan }^{-1}\left(\frac{\sqrt{1+\cos x}}{\sqrt{1-\cos x}}\right)$ in its simplest form.

Answer 1

${\tan }^{-1}\left(\frac{\sqrt{1+\cos x}}{\sqrt{1-\cos x}}\right)={\tan }^{-1}\left(\frac{\sqrt{2{\cos }^2\frac{x}{2}}}{\sqrt{2{\sin }^2\frac{x}{2}}}\right)={\tan }^{-1}\left(\sqrt{\frac{2{\cos }^2\frac{x}{2}}{2{\sin }^2\frac{x}{2}}}\right)={\tan }^{-1}\left(\sqrt{{\cot }^2\frac{x}{2}}\right)={\tan }^{-1}\left(\cot \frac{x}{2}\right)={\tan }^{-1}\left(\tan (\frac{\pi }{2}-\frac{x}{2})\right)=\frac{\pi }{2}-\frac{x}{2}$