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