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