\int_{x=0}^2\int_{y=0}^{2-x}\sin\left(\frac{\pi}{2}x\right)\cos\left(\frac{\pi}{2}y\right)\,dy\,dx&=&\int_{x=0}^2\sin\left(\frac{\pi}{2}x\right)\,dx\int_{y=0}^{2-x}\cos\left(\frac{\pi}{2}y\right)\,dy\\&=&\frac{2}{\pi}\int_{x=0}^2
\sin\left(\frac{\pi}{2}x\right)
\sin\left(\frac{\pi}{2}y\right)_{y=0}^{y=2-x}\,dx\\&=&\frac{2}{\pi}\int_{x=0}^{2}\sin\left(\frac{\pi}{2}x\right)\sin\left(\pi-\frac{\pi}{2}x\right)\,dx\\&=&\frac{2}{\pi}\int_{x=0}^{2}\sin\left(\frac{\pi}{2}x\right)\sin\left(\frac{\pi}{2}x\right)\,dx\\&=&\frac{2}{\pi}\cdot\frac{1}{2}\int_{x=0}^{2}(1-\cos
\pi x)\,dx\\&=&\frac{1}{\pi}\left(x-\frac{1}{\pi}\sin\pi
x\right)_{x=0}^{x=2}\\&=&\frac{2}{\pi}