Given the equation: \[ \sin^{-1} x + \cos^{-1} y = \frac{3\pi}{10}. \] Using the identity: \[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}, \] we can express \( \cos^{-1} y \) as: \[ \cos^{-1} y = \frac{\pi}{2} - \sin^{-1} y. \] Substituting this into the given equation yields: \[ \sin^{-1} x + \left( \frac{\pi}{2} - \sin^{-1} y \right) = \frac{3\pi}{10}. \] Simplifying the equation: \[ \sin^{-1} x + \frac{\pi}{2} - \sin^{-1} y = \frac{3\pi}{10}. \] Rearranging to isolate \( \sin^{-1} x - \sin^{-1} y \): \[ \sin^{-1} x - \sin^{-1} y = \frac{3\pi}{10} - \frac{\pi}{2}. \] Further simplification: \[ \sin^{-1} x - \sin^{-1} y = \frac{3\pi}{10} - \frac{5\pi}{10} = -\frac{2\pi}{10} = -\frac{\pi}{5}. \] Next, we need to calculate \( \cos^{-1} x + \sin^{-1} y \). We know that: \[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x. \] Substituting this expression: \[ \cos^{-1} x + \sin^{-1} y = \left( \frac{\pi}{2} - \sin^{-1} x \right) + \sin^{-1} y. \] Rearranging the terms: \[ \cos^{-1} x + \sin^{-1} y = \frac{\pi}{2} - (\sin^{-1} x - \sin^{-1} y). \] Substituting the previously found value of \( \sin^{-1} x - \sin^{-1} y = -\frac{\pi}{5} \): \[ \cos^{-1} x + \sin^{-1} y = \frac{\pi}{2} - \left(-\frac{\pi}{5}\right). \] Simplifying the expression: \[ \cos^{-1} x + \sin^{-1} y = \frac{\pi}{2} + \frac{\pi}{5}. \] Finding a common denominator: \[ \cos^{-1} x + \sin^{-1} y = \frac{5\pi}{10} + \frac{2\pi}{10} = \frac{7\pi}{10}. \]
Final Answer: \[ \boxed{\frac{7\pi}{10}} \]