Question

If \( \sin^{-1} x + \cos^{-1} y = \frac{3\pi}{10} \), then the value of \( \cos^{-1} x + \sin^{-1} y \) is:

• A. \( \frac{\pi}{10} \)
• B. \( \frac{7\pi}{10} \)
• C. \( \frac{9\pi}{10} \)
• D. \( \frac{3\pi}{10} \)

Hint:

Use trigonometric identities like \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \) to simplify mixed inverse trigonometric equations step by step.

Answer 1

The Correct Option is B

From the given equation: \[ \sin^{-1} x + \cos^{-1} y = \frac{3\pi}{10}. \] Using the identity: \[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}, \] we know: \[ \cos^{-1} y = \frac{\pi}{2} - \sin^{-1} y. \] Substitute \( \cos^{-1} y = \frac{\pi}{2} - \sin^{-1} y \) into the equation: \[ \sin^{-1} x + \left( \frac{\pi}{2} - \sin^{-1} y \right) = \frac{3\pi}{10}. \] Simplify: \[ \sin^{-1} x + \frac{\pi}{2} - \sin^{-1} y = \frac{3\pi}{10}. \] Rearrange to find \( \sin^{-1} x - \sin^{-1} y \): \[ \sin^{-1} x - \sin^{-1} y = \frac{3\pi}{10} - \frac{\pi}{2}. \] Simplify: \[ \sin^{-1} x - \sin^{-1} y = \frac{3\pi}{10} - \frac{5\pi}{10} = -\frac{2\pi}{10} = -\frac{\pi}{5}. \] Now, calculate \( \cos^{-1} x + \sin^{-1} y \): \[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x. \] Substitute: \[ \cos^{-1} x + \sin^{-1} y = \left( \frac{\pi}{2} - \sin^{-1} x \right) + \sin^{-1} y. \] Simplify: \[ \cos^{-1} x + \sin^{-1} y = \frac{\pi}{2} - (\sin^{-1} x - \sin^{-1} y). \] Substitute \( \sin^{-1} x - \sin^{-1} y = -\frac{\pi}{5} \): \[ \cos^{-1} x + \sin^{-1} y = \frac{\pi}{2} - \left(-\frac{\pi}{5}\right). \] Simplify: \[ \cos^{-1} x + \sin^{-1} y = \frac{\pi}{2} + \frac{\pi}{5}. \] Convert to a common denominator: \[ \cos^{-1} x + \sin^{-1} y = \frac{5\pi}{10} + \frac{2\pi}{10} = \frac{7\pi}{10}. \] 

 Final Answer: \[ \boxed{\frac{7\pi}{10}} \]