Question

Find the value of $\tan \left[\frac{1}{2}{\cos }^{-1}\frac{\sqrt{5}}{3}\right]$

Answer 1

Given,

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

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

$\Rightarrow \cos x=\frac{\sqrt{5}}{3}$

$\therefore y=\tan \frac{1}{2}x$

$y=\tan \frac{x}{2}$

$y=\sqrt{\frac{1-\cos x}{1+\cos x}}$

$=\sqrt{\frac{1-\frac{\sqrt{5}}{3}}{1+\frac{\sqrt{5}}{3}}}$

$=\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}}$

rationalizing the factor, we get,

$y=\frac{3-\sqrt{5}}{2}$