## Precalculus (6th Edition) Blitzer

The given expression on the left side $\tan \frac{x}{2}\left( 1+\cos x \right)$ can be simplified by using the half-angle formula. As per the half-angle formula, $\tan \frac{x}{2}=\frac{\sin x}{1+\cos x}$. \begin{align} & \tan \frac{x}{2}\left( 1+\cos x \right)=\frac{\sin x}{1+\cos x}.\left( 1+\cos x \right) \\ & =\sin x \end{align} Hence, the expression on the left-side is equal to the expression on the right-side.