Question:
Write the value of $\cos ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right)$.
Solution:
Let $y=\cos ^{-1}\left(\frac{3}{5}\right)$
$\Rightarrow \cos y=\frac{3}{5}$
Now,
$\cos ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right)=\cos ^{2}\left(\frac{1}{2} y\right)$
$=\frac{\cos y+1}{2} \quad\left[\because \cos 2 x=2 \cos ^{2} x-1\right]$
$=\frac{\frac{3}{5}+1}{2}$
$=\frac{\frac{8}{5}}{2}$
$=\frac{4}{5}$
$\therefore \cos ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right)=\frac{4}{5}$
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