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