Answer
See the explanation below.
Work Step by Step
The expression on the left side ${{\sin }^{2}}\frac{t}{2}$ can be further expressed as ${{\left( \sin \frac{t}{2} \right)}^{2}}$.
${{\sin }^{2}}\frac{t}{2}={{\left( \sin \frac{t}{2} \right)}^{2}}$
And the above expression can be further simplified by using the power reducing formula ${{\sin }^{2}}x=\frac{1-\cos 2x}{2}$. Thus, the above expression can be further simplified as:
$\begin{align}
& {{\left( \sin \frac{t}{2} \right)}^{2}}=\left( \sqrt{\frac{1-\cos t}{2}} \right) \\
& =\frac{1-\cos t}{2}
\end{align}$
Then, multiplying and dividing the expression by tan t.
$\begin{align}
& \frac{1-\cos t}{2}=\frac{1-\cos t}{2}.\frac{\tan t}{\tan t} \\
& =\frac{\tan t-\cos t.\tan t}{2\tan t}
\end{align}$
Now, apply the quotient identity $\tan t=\frac{\sin t}{\cos t}$
$\begin{align}
& \frac{\tan t-\cos t.\tan t}{2\tan t}=\frac{\tan t-\cos t.\frac{\sin t}{\cos t}}{2\tan t} \\
& =\frac{\tan t-\sin t}{2\tan t}
\end{align}$
Hence, the left side is equal to the right side, ${{\sin }^{2}}\frac{t}{2}=\frac{\tan t-\sin t}{2\tan t}$.
