Answer
The value is $\frac{1}{5}$.
Work Step by Step
We have to compute the value in the following manner:
Now, let us consider,
$\begin{align}
& \theta ={{\cos }^{-1}}\frac{3}{5} \\
& \cos \theta =\frac{3}{5}
\end{align}$
Then, solve the equation:
$\begin{align}
& {{\sin }^{2}}\frac{1}{2}{{\cos }^{-1}}\frac{3}{5}={{\sin }^{2}}\frac{1}{2}\theta \\
& =\frac{1-\cos 2\times \frac{1}{2}\theta }{2} \\
& =\frac{1-\cos \theta }{2}
\end{align}$
By further simplify the equation:
$\begin{align}
& {{\sin }^{2}}\frac{1}{2}{{\cos }^{-1}}\frac{3}{5}=\frac{1-\frac{3}{5}}{2} \\
& =\frac{\frac{2}{5}}{2} \\
& =\frac{1}{5}
\end{align}$
The value is $\frac{1}{5}$.
