Mark the tick against the correct answer in the following:
Question:

Mark the tick against the correct answer in the following:

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

A. $\frac{1}{\sqrt{5}}$

B. $\frac{2}{\sqrt{5}}$

C. $\frac{1}{\sqrt{10}}$

D. $\frac{2}{\sqrt{10}}$

Solution:

To Find: The value of $\sin \left(\frac{1}{2} \cos ^{-1} \frac{4}{5}\right)$

Let $x=\cos ^{-1} \frac{4}{5}$

$\Rightarrow \cos x=\frac{4}{5}$

Therefore $\sin \left(\frac{1}{2} \cos ^{-1} \frac{4}{5}\right)$ becomes $\sin \left(\frac{1}{2} x\right), i . e \sin \left(\frac{x}{2}\right)$

We know that $\sin \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos x}{2}}$

$=\sqrt{\frac{1-\frac{4}{5}}{2}}$

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

$\sin \left(\frac{x}{2}\right)=\frac{1}{\sqrt{10}}$