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. $5 - \sqrt{2}$

Answer 1

Answer: (B)

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}} $
$= \sqrt{\left(\sqrt{5} -2\right)^{2}}=\sqrt{5}-2$