Question:
The value of the expression $\tan \left(\frac{1}{2} \cos ^{-1} \frac{2}{\sqrt{5}}\right)$ is
(a) $2+\sqrt{5}$
(b) $\sqrt{5}-2$
(c) $\frac{\sqrt{5}+2}{2}$
(d) $5+\sqrt{2}$
Solution:
Let $\cos ^{-1} \frac{2}{\sqrt{5}}=\theta$. Then,
$\cos \theta=\frac{2}{\sqrt{5}}$
Now,
$\tan \left(\frac{1}{2} \cos ^{-1} \frac{2}{\sqrt{5}}\right)$
$=\tan \left(\frac{\theta}{2}\right)$
$=\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}$
$=\sqrt{\frac{1-\frac{2}{\sqrt{5}}}{1+\frac{2}{\sqrt{5}}}}$
$=\sqrt{\frac{\sqrt{5}-2}{\sqrt{5}+2}}$
$=\sqrt{\frac{(\sqrt{5}-2)^{2}}{(\sqrt{5}+2)(\sqrt{5}-2)}}$
$=\sqrt{\frac{(\sqrt{5}-2)^{2}}{5-4}}$
$=\sqrt{5}-2$
Thus, the value of the given expression is $\sqrt{5}-2$.
Hence, the correct answer is option (b).