Q. The value of the expression $\tan \left( \frac{1}{2} \cos^{-1} \frac{2}{\sqrt{5}} \right)$ is

Solution:

Let $\theta =\cos^{-1} \frac{2}{\sqrt{5}} \Rightarrow \cos\theta = \frac{2}{\sqrt{5}}$
$ \therefore \tan\left(\frac{1}{2} \cos^{-1} \frac{2}{\sqrt{5}}\right)= \tan \frac{\theta}{2} = \sqrt{\frac{1-\cos\theta}{1+\cos\theta}} $
$= \sqrt{\frac{1- \frac{2}{\sqrt{5}}}{1+ \frac{2}{\sqrt{5}}}} = \sqrt{\frac{\sqrt{5} -2}{\sqrt{5 } +2}} = \sqrt{\left(\sqrt{5} -2\right)^{2}}=\sqrt{5}-2$