Question

If \( \sin^{-1}x+\sin^{-1}y=\dfrac{2\pi}{3} \), then \( \cos^{-1}x+\cos^{-1}y \) is equal to

• A. \( -\dfrac{\pi}{2} \)
• B. \( \dfrac{\pi}{2} \)
• C. \( \pi \)
• D. \( \dfrac{2\pi}{3} \)
• E. \( \dfrac{\pi}{3} \)

Hint:

Remember the identity \( \sin^{-1}t+\cos^{-1}t=\frac{\pi}{2} \). It directly converts sums of inverse sine into sums of inverse cosine and vice versa.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem uses the complementary property of inverse sine and inverse cosine functions.
Step 2: Key Formula or Approach:
For any value of \(t\) in the domain \([-1, 1]\), the following identity holds:
\[ \sin^{-1}(t) + \cos^{-1}(t) = \frac{\pi}{2} \] We can apply this identity to both \(x\) and \(y\).
Step 3: Detailed Explanation:
We are given the equation:
\[ \sin^{-1}x + \sin^{-1}y = \frac{2\pi}{3} \quad \cdots (1) \] We want to find the value of:
\[ S = \cos^{-1}x + \cos^{-1}y \] From the complementary identity, we can write:
\[ \sin^{-1}x = \frac{\pi}{2} - \cos^{-1}x \] \[ \sin^{-1}y = \frac{\pi}{2} - \cos^{-1}y \] Substitute these expressions into the given equation (1):
\[ (\frac{\pi}{2} - \cos^{-1}x) + (\frac{\pi}{2} - \cos^{-1}y) = \frac{2\pi}{3} \] Combine the constants:
\[ \pi - (\cos^{-1}x + \cos^{-1}y) = \frac{2\pi}{3} \] Substitute \(S\) for \( \cos^{-1}x + \cos^{-1}y \):
\[ \pi - S = \frac{2\pi}{3} \] Now, solve for S:
\[ S = \pi - \frac{2\pi}{3} \] \[ S = \frac{3\pi - 2\pi}{3} = \frac{\pi}{3} \] Step 4: Final Answer:
The value of \( \cos^{-1}x + \cos^{-1}y \) is \(\frac{\pi}{3}\). Therefore, option (E) is the correct answer.