Answer
$\dfrac{dy}{dx}=\dfrac{\cos{\pi x}}{\sin{\pi y}}.$
Work Step by Step
$\dfrac{d}{dx}((\sin{\pi x}+\cos{\pi y})^2)=\dfrac{d}{dx}(2)\rightarrow$
Using the Chain Rule with $u=\sin{\pi x}+\cos{\pi y}\rightarrow\dfrac{du}{dx}=\pi\cos{\pi x}-\pi\sin{\pi y}\rightarrow$
$\dfrac{d}{dx}((\sin{\pi x}+\cos{\pi y})^2)$
$=2(\sin{\pi x}+\cos{\pi y})(\pi\cos{\pi x}-\dfrac{dy}{dx}(\pi\sin{\pi y)})$
$2(\sin{\pi x}+\cos{\pi y})(\pi\cos{\pi x}-\dfrac{dy}{dx}(\pi\sin{\pi y)})=0\rightarrow$
$\dfrac{dy}{dx}(\pi\sin{\pi y})=\pi\cos{\pi x}\rightarrow$
$\dfrac{dy}{dx}=\dfrac{\cos{\pi x}}{\sin{\pi y}}.$
