Question
Evaluate:
\[ \tan^{-1}\left\{2\sin\left(2\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\right)\right\} \]
Solution
First, evaluate:
\[ \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6} \]
So,
\[ 2\cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \]
Now,
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Thus,
\[ 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]
Now evaluate:
\[ \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \]
Final Answer:
\[ \boxed{\frac{\pi}{3}} \]
Key Concept
Break the expression step-by-step using standard values and identities.