Explanation

(b)
$\sin\left( \frac{1}{2}\cos^{- 1}\frac{4}{5} \right)$
Now, put $\frac{4}{5} = \cos{2\theta}$
$\therefore\ \sin\left( \frac{1}{2} \times 2\theta \right)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sin{\theta = \sqrt{\frac{1 - \cos{2\theta}}{2}}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sqrt{\frac{1 - \frac{4}{5}}{2}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sqrt{\frac{1}{5 \times 2}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{1}{\sqrt{10}}$