Question

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

• A. \( 1 - y^2 \)
• B. \( y^2 \)
• C. \( (1 - y)^2 \)
• D. \( (1 - \sqrt{y})^2 \)

Hint:

When \( \sin^{-1} x + \sin^{-1} y = \frac{\pi}{2} \), use the identity \( x = \sqrt{1 - y^2} \) to find the relationship between \( x^2 \) and \( y^2 \).

Answer 1

The Correct Option is A

Step 1: Using the given equation.
We are given that: \[ \sin^{-1} x + \sin^{-1} y = \frac{\pi}{2} \] From the identity \( \sin^{-1} x + \sin^{-1} y = \frac{\pi}{2} \), we know that: \[ \sin^{-1} x = \frac{\pi}{2} - \sin^{-1} y \] Therefore, \( x = \cos(\sin^{-1} y) \).
Step 2: Using the trigonometric identity.
Using the identity \( \cos(\sin^{-1} y) = \sqrt{1 - y^2} \), we have: \[ x = \sqrt{1 - y^2} \] Squaring both sides, we get: \[ x^2 = 1 - y^2 \]
Step 3: Conclusion.
Thus, \( x^2 = 1 - y^2 \). The correct answer is (1) \( 1 - y^2 \).