Question

If $4\sin^{-1}x + \cos^{-1}x = \pi$, then $x$ is equal to

• A. $\frac{1}{2}$
• B. $2$
• C. $1$
• D. $\frac{1}{3}$

Hint:

$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$ for $x \in [-1,1]$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We know $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$ for $x \in [-1,1]$.
Step 2: Detailed Explanation:
Let $\sin^{-1}x = \theta$, then $\cos^{-1}x = \frac{\pi}{2} - \theta$.
Substitute: $4\theta + \left(\frac{\pi}{2} - \theta\right) = \pi \Rightarrow 3\theta + \frac{\pi}{2} = \pi \Rightarrow 3\theta = \frac{\pi}{2} \Rightarrow \theta = \frac{\pi}{6}$.
So $\sin^{-1}x = \frac{\pi}{6} \Rightarrow x = \sin\frac{\pi}{6} = \frac{1}{2}$.
Step 3: Final Answer:
$x = \frac{1}{2}$.