Question
If
\[ \sin^{-1}\left(\frac{1}{3}\right) + \cos^{-1}x = \frac{\pi}{2} \]
Find \( x \).
Solution
We use identity:
\[ \sin^{-1}a + \cos^{-1}a = \frac{\pi}{2} \]
Comparing,
\[ \cos^{-1}x = \frac{\pi}{2} – \sin^{-1}\left(\frac{1}{3}\right) \]
\[ = \cos^{-1}\left(\frac{1}{3}\right) \]
Thus,
\[ x = \frac{1}{3} \]
Final Answer:
\[ \boxed{\frac{1}{3}} \]
Key Concept
Use the identity \( \sin^{-1}a + \cos^{-1}a = \frac{\pi}{2} \) to compare expressions directly.