Find the value of (1 2 ^ -1 2 5 ). - Vidyadip

Question

Find the value of $\tan \left(\frac{1}{2} \cos ^{-1} \frac{2}{\sqrt{5}}\right)$.

✓

Answer

Suppose $\cos ^{-1} \frac{2}{\sqrt{5}}=\theta$ then $\cos \theta=\frac{2}{\sqrt{5}}$$
\begin{aligned}
\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{1-2 / 55}{1+2 / 55}}=\sqrt{\frac{\sqrt{5}-2}{\sqrt{5}+2}} \\
& =\sqrt{\frac{(\sqrt{5}-2)^2}{5-4}}=\sqrt{5}-2 \text {}
\end{aligned}
$

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