Question
Find the value of:
\[ \cos\left(2\sin^{-1}\left(\frac{1}{3}\right)\right) \]
Solution
Let
\[ \theta = \sin^{-1}\left(\frac{1}{3}\right) \]
Then,
\[ \sin \theta = \frac{1}{3} \]
Using identity:
\[ \cos 2\theta = 1 – 2\sin^2\theta \]
Substitute:
\[ \cos 2\theta = 1 – 2\left(\frac{1}{3}\right)^2 \]
\[ = 1 – 2 \cdot \frac{1}{9} = 1 – \frac{2}{9} = \frac{7}{9} \]
Final Answer:
\[ \boxed{\frac{7}{9}} \]
Key Concept
Use the identity \( \cos 2\theta = 1 – 2\sin^2\theta \) to simplify expressions involving inverse sine.