1. The problem is to find the value of $y$ given the equation $(\sin x + \cos x) y = \cos^2 x$ at $x = \frac{\pi}{2}$. 2. Substitute $x = \frac{\pi}{2}$ into the equation: $$(\sin \frac{\pi}{2} + \cos \frac{\pi}{2}) y = \cos^2 \frac{\pi}{2}$$ 3. Evaluate the trigonometric functions: $\sin \frac{\pi}{2} = 1$ and $\cos \frac{\pi}{2} = 0$ 4. Substitute these values back into the equation: $$(1 + 0) y = 0^2$$ which simplifies to $$y = 0$$ 5. Therefore, the value of $y$ at $x = \frac{\pi}{2}$ is $0$.