Answer
See the explanation below.
Work Step by Step
Let us consider the right side of the given expression:
$\frac{1+\cos x}{\sin x}$
By using the trigonometric identity $\tan \frac{x}{2}=\frac{\sin x}{1+\cos x}$, the above expression can be further simplified by multiplying the numerator and denominator by $\frac{1}{1+\cos x}$:
$\begin{align}
& \frac{1+\cos x}{\sin x}=\frac{\left( 1+\cos x \right)\times \frac{1}{1+\cos x}}{\sin x\times \frac{1}{1+\cos x}} \\
& =\frac{\frac{1+\cos x}{1+\cos x}}{\frac{\sin x}{1+\cos x}} \\
& =\frac{1}{\tan \frac{x}{2}} \\
& =\cot \frac{x}{2}
\end{align}$
Hence, the left side of the given expression is equal to the right side, which is $\cot \frac{x}{2}=\frac{1+\cos x}{\sin x}$.
