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

The correct option is B $\sqrt{5}-2$

$\text{Let}{\cos }^{-1}\frac{2}{\sqrt{5}}=x\text{i.e.},\cos x=\frac{2}{\sqrt{5}}$
$\text{Then},\tan \left(\frac{1}{2}{\cos }^{-1}\frac{2}{\sqrt{5}}\right)=\tan \frac{x}{2}$
$=\frac{\sin x}{1+\cos x}$

$=\frac{\sqrt{1-\frac{4}{5}}}{1+\frac{2}{\sqrt{5}}}=\sqrt{5}-2$