Write the value

Let $y=\cos ^{-1}\left(\frac{3}{5}\right)$

$\Rightarrow \cos y=\frac{3}{5}$

Now,

$\cos ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right)=\cos ^{2}\left(\frac{1}{2} y\right)$

$=\frac{\cos y+1}{2} \quad\left[\because \cos 2 x=2 \cos ^{2} x-1\right]$

$=\frac{\frac{3}{5}+1}{2}$

$=\frac{\frac{8}{5}}{2}$

$=\frac{4}{5}$

$\therefore \cos ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right)=\frac{4}{5}$