\(\sin^2 x + \sin^2 y = 1\) i.e., \(\sin^2 x = 1 - \sin^2 y\) i.e., \(\sin^2 x = \cos^2 y\) i.e., \(\sin x = \cos y\) i.e., \(\sin x = \sin(90^\circ - y)\) i.e., \(x = 90^\circ - y\) i.e., \(x + y = 90^\circ\) i.e., \(\frac{x + y}{2} = 45^\circ\) \(\therefore \sin \frac{x + y}{2} + \cos \frac{x + y}{2}\) \(= \sin 45^\circ + \cos 45^\circ\) \(= \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}\) \(= \frac{2}{\sqrt{2}}\) \(= \sqrt{2}\)