Question
Evaluate:
\[ \sin^{-1}\left(\frac{1}{3}\right) – \cos^{-1}\left(-\frac{1}{3}\right) \]
Solution
We use identity:
\[ \cos^{-1}(-x) = \pi – \cos^{-1}(x) \]
So,
\[ \cos^{-1}\left(-\frac{1}{3}\right) = \pi – \cos^{-1}\left(\frac{1}{3}\right) \]
Now substitute:
\[ \sin^{-1}\left(\frac{1}{3}\right) – \left(\pi – \cos^{-1}\left(\frac{1}{3}\right)\right) \]
\[ = \sin^{-1}\left(\frac{1}{3}\right) + \cos^{-1}\left(\frac{1}{3}\right) – \pi \]
Using identity:
\[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \]
Thus,
\[ = \frac{\pi}{2} – \pi = -\frac{\pi}{2} \]
Final Answer:
\[ \boxed{-\frac{\pi}{2}} \]
Key Concept
Use identities \( \cos^{-1}(-x) \) and \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \) to simplify expressions.