Answer
See the full 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 $\frac{1}{\cos x}=\sec x$,
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 numerator and denominator by $\frac{1}{\cos \theta }$, we get:
$\begin{align}
& \frac{1+\cos \theta }{2}=\frac{\left( 1+\cos \theta \right)\times \frac{1}{\cos \theta }}{2\times \frac{1}{\cos \theta }} \\
& =\frac{\frac{1}{\cos \theta }+\frac{\cos \theta }{\cos \theta }}{2\frac{1}{\cos \theta }} \\
& =\frac{\sec \theta +1}{2\sec \theta }
\end{align}$
Hence, the left side of the given expression is equal to the right side, which is ${{\cos }^{2}}\frac{\theta }{2}=\frac{\sec \theta +1}{2\sec \theta }$.
