Question 15.5.4: Prove that 1+cos x/1-cos x=(cosec x+cot x)² ....

\begin{aligned}\text { RHS } & =(\operatorname{cosec} x+\cot x)^{2} \\  \\& =\left(\frac{1}{\sin x}+\frac{\cos x}{\sin x}\right)^{2}=\left(\frac{1+\cos x}{\sin x}\right)^{2}\\  \\& =\left\{\frac{2 \cos ^{2}\left(\frac{x}{2}\right)}{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}\right\}^{2}=\left\{\frac{\cos \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)}\right\}^{2}=\cot ^{2} \frac{x}{2} .\end{aligned}\\  \\ \therefore \quad LHS = RHS.