Question
If
\[ 4\cos^{-1}x + \sin^{-1}x = \pi \]
Find \( x \).
Solution
Use identity:
\[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \Rightarrow \sin^{-1}x = \frac{\pi}{2} – \cos^{-1}x \]
Substitute:
\[ 4\cos^{-1}x + \left(\frac{\pi}{2} – \cos^{-1}x\right) = \pi \]
\[ 3\cos^{-1}x + \frac{\pi}{2} = \pi \]
\[ 3\cos^{-1}x = \frac{\pi}{2} \]
\[ \cos^{-1}x = \frac{\pi}{6} \]
Thus,
\[ x = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]
Final Answer:
\[ \boxed{\frac{\sqrt{3}}{2}} \]
Key Concept
Use identity \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \) to reduce equation.