cot inverse [ root ( 1 + sinx ) + root ( 1 - sinx ) / root ( 1 + sinx ) - root ( 1 - sinx ) ] = x/2

Hi Manonarayanan,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}.={\cot }^{-1}\left(\frac{\sqrt{1+\sin \mathrm{x}}+\sqrt{1-\sin \mathrm{x}}}{\sqrt{1+\sin \mathrm{x}}-\sqrt{1-\sin \mathrm{x}}}\right) \\ {\cos }^2\frac{\mathrm{\alpha }}{2}+{\sin }^2\frac{\mathrm{\alpha }}{2}=1 \mathrm{and} \sin \mathrm{\alpha }=2\sin \frac{\mathrm{\alpha }}{2}.\cos \frac{\mathrm{\alpha }}{2}, \mathrm{we} \mathrm{get} \\ \mathrm{L}.\mathrm{H}.\mathrm{S}.={\cot }^{-1}\left(\frac{\sqrt{{\cos }^2\frac{\mathrm{x}}{2}+{\sin }^2\frac{\mathrm{x}}{2}+2\sin \frac{\mathrm{x}}{2}.\cos \frac{\mathrm{x}}{2}}+\sqrt{{\cos }^2\frac{\mathrm{x}}{2}+{\sin }^2\frac{\mathrm{x}}{2}-2\sin \frac{\mathrm{x}}{2}.\cos \frac{\mathrm{x}}{2}}}{\sqrt{{\cos }^2\frac{\mathrm{x}}{2}+{\sin }^2\frac{\mathrm{x}}{2}+2\sin \frac{\mathrm{x}}{2}.\cos \frac{\mathrm{x}}{2}}-\sqrt{{\cos }^2\frac{\mathrm{x}}{2}+{\sin }^2\frac{\mathrm{x}}{2}-2\sin \frac{\mathrm{x}}{2}.\cos \frac{\mathrm{x}}{2}}}\right) \\ \mathrm{Using} \mathrm{identity} {\left(\mathrm{a}\pm \mathrm{b}\right)}^2={\mathrm{a}}^2+{\mathrm{b}}^2\pm 2\mathrm{ab} , \mathrm{we} \mathrm{get}, \\ \mathrm{L}.\mathrm{H}.\mathrm{S}.={\cot }^{-1}\left(\frac{\sqrt{{\left(\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}\right)}^2}+\sqrt{{\left(\cos \frac{\mathrm{x}}{2}-\sin \frac{\mathrm{x}}{2}\right)}^2}}{\sqrt{{\left(\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}\right)}^2}-\sqrt{{\left(\cos \frac{\mathrm{x}}{2}-\sin \frac{\mathrm{x}}{2}\right)}^2}}\right) \\ ={\cot }^{-1}\left(\frac{\left(\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}\right)+\left(\cos \frac{\mathrm{x}}{2}-\sin \frac{\mathrm{x}}{2}\right)}{\left(\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}\right)-\left(\cos \frac{\mathrm{x}}{2}-\sin \frac{\mathrm{x}}{2}\right)}\right) \\ ={\cot }^{-1}\left(\frac{\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}+\cos \frac{\mathrm{x}}{2}-\sin \frac{\mathrm{x}}{2}}{\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}-\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}}\right) \\ ={\cot }^{-1}\left(\frac{2\cos \frac{\mathrm{x}}{2}}{2\sin \frac{\mathrm{x}}{2}}\right) \\ ={\cot }^{-1}\left(\frac{\cos \frac{\mathrm{x}}{2}}{\sin \frac{\mathrm{x}}{2}}\right) \\ ={\cot }^{-1}\left(\cot \frac{\mathrm{x}}{2}\right) \left(\mathrm{As}\frac{\cos \mathrm{\alpha }}{\sin \mathrm{\alpha }}=\cot \mathrm{\alpha }\right) \\ \mathbf{\Rightarrow }\mathbf{L}\mathbf{.}\mathbf{H}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{=}\frac{\mathbf{x}}{\mathbf{2}}\mathbf{=}\mathbf{R}\mathbf{.}\mathbf{H}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{ }\left(\mathbf{Hence}\mathbf{ }\mathbf{proved}\right) \end{gathered}$$

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