Question
If
\[ 4\sin^{-1}x + \cos^{-1}x = \pi \]
Find \( x \).
Solution
We use identity:
\[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \]
So,
\[ \cos^{-1}x = \frac{\pi}{2} – \sin^{-1}x \]
Substitute into given equation:
\[ 4\sin^{-1}x + \left(\frac{\pi}{2} – \sin^{-1}x\right) = \pi \]
\[ 3\sin^{-1}x + \frac{\pi}{2} = \pi \]
\[ 3\sin^{-1}x = \frac{\pi}{2} \]
\[ \sin^{-1}x = \frac{\pi}{6} \]
Therefore,
\[ x = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Final Answer:
\[ \boxed{\frac{1}{2}} \]
Key Concept
Use the identity \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \) to simplify equations efficiently.