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To Find: The value of $\tan \frac{1}{2}\left(\cos ^{-1} \frac{\sqrt{5}}{3}\right)$

Let, $x=\cos ^{-1} \frac{\sqrt{5}}{3}$

$\Rightarrow \cos x=\frac{\sqrt{5}}{3}$

Now, $\tan \frac{1}{2}\left(\cos ^{-1} \frac{\sqrt{5}}{3}\right)$ becomes

$\tan \frac{1}{2}\left(\cos ^{-1} \frac{\sqrt{5}}{3}\right)=\tan \frac{1}{2}(x)=\tan \frac{x}{2}$

$=\sqrt{\frac{1-\cos x}{1+\cos x}}$

$=\sqrt{\frac{1-\left(\frac{\sqrt{5}}{3}\right)}{1+\frac{\sqrt{5}}{2}}}$

$=\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}}$

$=\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}} \times \sqrt{\frac{3-\sqrt{5}}{3-\sqrt{5}}}$

$\tan \frac{1}{2}\left(\cos ^{-1} \frac{\sqrt{5}}{3}\right)=\frac{3-\sqrt{5}}{2}$