Q43 / 90 · JEE Main 2012 (19 May Online) · Mathematics
A value of $\tan ^{-1}\left(\sin \left(\cos ^{-1}\left(\sqrt{\frac{2}{3}}\right)\right)\right)$ is
- A$\frac{\pi}{4}$
- B$\frac{\pi}{2}$
- C$\frac{\pi}{3}$
- Correct answerD$\frac{\pi}{6}$
Solution
Consider $\tan ^{-1}\left[\sin \left(\cos ^{-1} \sqrt{\frac{2}{3}}\right)\right]$
$$
\begin{aligned}
& \text { Let } \cos ^{-1} \sqrt{\frac{2}{3}}=\theta \Rightarrow \cos \theta=\sqrt{\frac{2}{3}} \\
& \Rightarrow \quad \sin \theta=\sqrt{1-\cos ^2 \theta}=\sqrt{1-\frac{2}{3}}=\sqrt{\frac{1}{3}} \\
& \therefore \quad \tan ^{-1}\left[\sin \left(\cos ^{-1} \sqrt{\frac{2}{3}}\right)\right]=\tan ^{-1}[\sin \theta] \\
& =\tan ^{-1}\left[\sqrt{\frac{1}{3}}\right]=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right) \\
& =\frac{\pi}{6}
\end{aligned}
$$
Hence, the correct answer is option: 'D'.