Question

The value of the expression tan $\left(\frac{1}{2}{\cos }^{-1}\frac{2}{\sqrt{5}}\right)$ is
​(a) $2+\sqrt{5}\left(\mathrm{b}\right)\sqrt{5}-2\left(\mathrm{c}\right)\frac{\sqrt{5}+2}{2}\left(\mathrm{d}\right)5+\sqrt{2}$

Answer 1

Let ${\cos }^{-1}\frac{2}{\sqrt{5}}=\theta$. Then,
$\cos \theta =\frac{2}{\sqrt{5}}$

Now,

$$\begin{gathered} \tan \left(\frac{1}{2}{\cos }^{-1}\frac{2}{\sqrt{5}}\right) \\ =\tan \left(\frac{\theta }{2}\right) \\ =\sqrt{\frac{1-\cos \theta }{1+\cos \theta }} \end{gathered}$$
$$\begin{gathered} =\sqrt{\frac{1-{\displaystyle \frac{2}{\sqrt{5}}}}{1+{\displaystyle \frac{2}{\sqrt{5}}}}} \\ =\sqrt{\frac{\sqrt{5}-2}{\sqrt{5}+2}} \\ =\sqrt{\frac{{\left(\sqrt{5}-2\right)}^2}{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}} \end{gathered}$$
$$\begin{gathered} =\sqrt{\frac{{\left(\sqrt{5}-2\right)}^2}{5-4}} \\ =\sqrt{5}-2 \end{gathered}$$

Thus, the value of the given expression is $\sqrt{5}-2$.

Hence, the correct answer is option (b).