The value of the expression

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).