$\mathit{c}\mathit{o}{\mathit{t}}^{-1}\left[\frac{\sqrt{1+\mathit{s}\mathit{i}\mathit{n}\mathit{x}}+\sqrt{1-\mathit{s}\mathit{i}\mathit{n}\mathit{x}}}{\sqrt{1+\mathit{s}\mathit{i}\mathit{n}\mathit{x}}-\sqrt{1-\mathit{s}\mathit{i}\mathit{n}\mathit{x}}}\right]$
=
[Since, $\mathit{s}\mathit{i}{\mathit{n}}^2$ A + $\mathit{c}\mathit{o}{\mathit{s}}^2$ A = 1]
= [Since, sin2A = 2 sinA cosA]
=
= $\mathit{c}\mathit{o}{\mathit{t}}^{-1}\left(\frac{2\mathit{c}\mathit{o}\mathit{s}\frac{\mathit{x}}{2}}{2\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{x}}{2}}\right)$ = $\mathit{c}\mathit{o}{\mathit{t}}^{-1}\left(\mathit{c}\mathit{o}\mathit{t}\frac{\mathit{x}}{2}\right)$
= $\frac{\mathit{x}}{2}$
Hence proved.
OR
$\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}}{\mathit{y}}\right)$ - $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}-\mathit{y}}{\mathit{x}+\mathit{y}}\right)$
$\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}}{\mathit{y}}\right)$ - $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\frac{\mathit{x}}{\mathit{y}}-1}{\frac{\mathit{x}}{\mathit{y}}+1}\right)$
= $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}}{\mathit{y}}\right)$ - $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\frac{\mathit{x}}{\mathit{y}}-1}{1+\frac{\mathit{x}}{\mathit{y}}}\right)$
$\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}}{\mathit{y}}\right)$ - $\left[\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}}{\mathit{y}}\right)-\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(1\right)\right]$ [Since $\mathit{t}\mathit{a}{\mathit{n}}^{-1}$ a - $\mathit{t}\mathit{a}{\mathit{n}}^{-1}$ b = $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{a}-\mathit{b}}{1+\mathit{a}\mathit{b}}\right)$]
$\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}}{\mathit{y}}\right)$ - $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}}{\mathit{y}}\right)$ + $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(1\right)$
= $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(1\right)$ = $\frac{\mathit{\pi }}{4}$
Thus $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}}{\mathit{y}}\right)$ - $\mathit{t}\mathit{a}{\mathit{n}}^{-1}\left(\frac{\mathit{x}-\mathit{y}}{\mathit{x}+\mathit{y}}\right)$ = $\frac{\mathit{\pi }}{4}$