Evaluate:${\tan }^{-1}\left(\frac{\sqrt{1+\cos x}-\sqrt{1-\cos x}}{\sqrt{1+\cos x}+\sqrt{1-\cos x}}\right)$   

We have,

${\tan }^{-1}\left(\frac{\sqrt{1+\cos x}-\sqrt{1-\cos x}}{\sqrt{1+\cos x}+\sqrt{1-\cos x}}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{\sqrt{1+2{\cos }^2\frac{x}{2}-1}-\sqrt{1-1+2si{n}^2\frac{x}{2}}}{\sqrt{1+2{\cos }^2\frac{x}{2}-1}+\sqrt{1-1+2si{n}^2\frac{x}{2}}}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{\sqrt{2{\cos }^2\frac{x}{2}}-\sqrt{2si{n}^2\frac{x}{2}}}{\sqrt{2{\cos }^2\frac{x}{2}}+\sqrt{2si{n}^2\frac{x}{2}}}\right)$

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

$\Rightarrow {\tan }^{-1}\left(\frac{\cos \frac{x}{2}-sin\frac{x}{2}}{\cos \frac{x}{2}+sin\frac{x}{2}}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{\cos \frac{x}{2}-sin\frac{x}{2}}{\cos \frac{x}{2}+sin\frac{x}{2}}\right)\times \left(\frac{\cos \frac{x}{2}-sin\frac{x}{2}}{\cos \frac{x}{2}-sin\frac{x}{2}}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{{\left(\cos \frac{x}{2}-sin\frac{x}{2}\right)}^2}{{\cos }^2\frac{x}{2}-si{n}^2\frac{x}{2}}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{\left({\cos }^2\frac{x}{2}+si{n}^2\frac{x}{2}-2sin\frac{x}{2}\cos \frac{x}{2}\right)}{{\cos }^2\frac{x}{2}-si{n}^2\frac{x}{2}}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{1-\sin x}{\cos x}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{1}{\cos x}-\frac{\sin x}{\cos x}\right)$

$\Rightarrow {\tan }^{-1}\left(\sec x-\tan x\right)$