Answer
The value is $\frac{9}{10}$.
Work Step by Step
Let us assume,
$\begin{align}
& {{\sin }^{-1}}\left( \frac{3}{5} \right)=x \\
& \sin x=\left( \frac{3}{5} \right)
\end{align}$
And the sine is computed by dividing the perpendicular and hypotenuse. So, the perpendicular (y) is 3 and the hypotenuse (r) is 5. Compute the base (x) as follows:
$\begin{align}
& {{x}^{2}}={{r}^{2}}-{{y}^{2}} \\
& ={{5}^{2}}-{{3}^{2}} \\
& =25-9
\end{align}$
Therefore, the value is
$\begin{align}
& x=\sqrt{16} \\
& =4
\end{align}$
So, the base is 4.
And compute the value as follows:
$\begin{align}
& {{\cos }^{2}}\left[ \frac{1}{2}{{\sin }^{-1}}\left( \frac{3}{5} \right) \right]={{\cos }^{2}}\frac{1}{2}\theta \\
& =\frac{1+\cos 2\times \frac{1}{2}\theta }{2} \\
& =\frac{1+\cos \theta }{2} \\
& =\frac{1+\frac{4}{5}}{2}
\end{align}$
By simplifying the equation, the result will be
$\begin{align}
& {{\cos }^{2}}\left[ \frac{1}{2}{{\sin }^{-1}}\left( \frac{3}{5} \right) \right]=\frac{\frac{9}{5}}{2} \\
& =\frac{9}{10}
\end{align}$
So, ${{\cos }^{2}}\left[ \frac{1}{2}{{\sin }^{-1}}\left( \frac{3}{5} \right) \right]=\frac{9}{10}$.
