Question
Find the value of:
\[ \cos\left(2\sin^{-1}\left(\frac{1}{2}\right)\right) \]
Solution
Let
\[ \theta = \sin^{-1}\left(\frac{1}{2}\right) \]
Then,
\[ \sin \theta = \frac{1}{2} \Rightarrow \theta = \frac{\pi}{6} \]
Now,
\[ \cos(2\theta) = \cos\left(2 \cdot \frac{\pi}{6}\right) = \cos\left(\frac{\pi}{3}\right) \]
\[ = \frac{1}{2} \]
Final Answer:
\[ \boxed{\frac{1}{2}} \]
Key Concept
Use known standard values to simplify inverse trigonometric expressions quickly.