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