Answer
$0$
Work Step by Step
$\sin{45^\circ}+\sin{135^\circ}+\sin{225^\circ}+\sin{315^\circ}=\sin{\frac{\pi}{4}}+\sin{\frac{3\pi}{5}}+\sin{\frac{5\pi}{4}}+\sin{\frac{7\pi}{4}}=\frac{\sqrt2}{2}+\frac{\sqrt2}{2}-\frac{\sqrt2}{2}-\frac{\sqrt2}{2}=0$.
