Question

.If π < x < 2π, prove that √1 + cos x + √1 − cos x / √1 + cos x – √1 − cos x = cot ( π /4 + x /2 )

Answer 1

$$\begin{gathered} \frac{\sqrt{1+\cos \mathrm{x}}+\sqrt{1-\cos \mathrm{x}}}{\sqrt{1+\cos \mathrm{x}}-\sqrt{1-\cos \mathrm{x}}} \\ =\frac{\sqrt{2{\cos }^2\left(\mathrm{x}/2\right)}{\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle \sqrt{2{\sin }^2\left(\mathrm{x}/2\right)}}}{\sqrt{2{\cos }^2\left(\mathrm{x}/2\right)}-\sqrt{2{\sin }^2\left(\mathrm{x}/2\right)}} \\ =\frac{{\displaystyle \sqrt{{\cos }^2\left(\mathrm{x}/2\right)}+\sqrt{{\sin }^2\left(\mathrm{x}/2\right)}}}{\sqrt{{\cos }^2\left(\mathrm{x}/2\right)}-\sqrt{{\sin }^2\left(\mathrm{x}/2\right)}} \\ =\frac{-\cos \left(\mathrm{x}/2\right)+\sin \left(\mathrm{x}/2\right)}{-\cos \left(\mathrm{x}/2\right)-\sin \left(\mathrm{x}/2\right)}\mathbf{}\mathbf{}\mathbf{}\left[\mathbf{as}\mathbf{,}\mathbf{}\mathbf{\pi }\mathbf{}\mathbf{<}\mathbf{}\mathbf{x}\mathbf{}\mathbf{<}\mathbf{\hspace{0.17em}}\mathbf{2}\mathbf{\pi }\mathbf{,}\mathbf{}\mathbf{so}\mathbf{}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{}\mathbf{<}\mathbf{}\frac{\mathbf{x}}{\mathbf{2}}\mathbf{}\mathbf{<}\mathbf{}\mathbf{\pi }\mathbf{}\mathbf{and}\mathbf{}\mathbf{sin}\mathbf{}\left(\frac{\mathbf{x}}{\mathbf{2}}\right)\mathbf{}\mathbf{>}\mathbf{}\mathbf{0}\mathbf{}\mathbf{and}\mathbf{}\mathbf{cos}\mathbf{}\left(\frac{\mathbf{x}}{\mathbf{2}}\right)\mathbf{}\mathbf{<}\mathbf{}\mathbf{0}\right] \\ =\frac{{\displaystyle \cos \left(\mathrm{x}/2\right)-\sin \left(\mathrm{x}/2\right)}}{{\displaystyle \cos \left(\mathrm{x}/2\right)+\sin \left(\mathrm{x}/2\right)}} \\ =\frac{{\displaystyle \frac{\cos {\displaystyle \left(\mathrm{x}/2\right)}}{{\displaystyle \sin \left(\mathrm{x}/2\right)}}-\frac{\sin \left(\mathrm{x}/2\right)}{\sin \left(\mathrm{x}/2\right)}}}{{\displaystyle \frac{\cos \left(\mathrm{x}/2\right)}{\sin \left(\mathrm{x}/2\right)}+\frac{\sin \left(\mathrm{x}/2\right)}{\sin \left(\mathrm{x}/2\right)}}} \\ =\frac{\cot \left(\mathrm{x}/2\right)-1}{\cot \left(\mathrm{x}/2\right)+1} \\ =\frac{{\displaystyle \cot \left(\mathrm{x}/2\right).\cot \left(\mathrm{\pi }/4\right)-1}}{{\displaystyle \cot \left(\mathrm{x}/2\right)+\cot \left(\mathrm{\pi }/4\right)}} \\ =\cot \left(\frac{\mathrm{\pi }}{4}+\frac{\mathrm{x}}{2}\right) \end{gathered}$$