Question
If
\[ \cos\left(\sin^{-1}\left(\frac{2}{5}\right) + \cos^{-1}x\right) = 0 \]
Find \( x \).
Solution
We know:
\[ \cos \theta = 0 \Rightarrow \theta = \frac{\pi}{2} \quad \text{(principal value)} \]
So,
\[ \sin^{-1}\left(\frac{2}{5}\right) + \cos^{-1}x = \frac{\pi}{2} \]
Using identity:
\[ \sin^{-1}a + \cos^{-1}a = \frac{\pi}{2} \]
Comparing,
\[ \cos^{-1}x = \frac{\pi}{2} – \sin^{-1}\left(\frac{2}{5}\right) = \cos^{-1}\left(\frac{2}{5}\right) \]
Thus,
\[ x = \frac{2}{5} \]
Final Answer:
\[ \boxed{\frac{2}{5}} \]
Key Concept
Use identity \( \sin^{-1}a + \cos^{-1}a = \frac{\pi}{2} \) to simplify equations directly.