### Instruction: Simplify $\sin (x - y) \cos y + \cos (x - y) \sin y.$.### Response: Ok, so we have $\sin (x - y) \cos y + \cos (x - y) \sin y$
Notice that $\sin (x - y) \cos y + \cos (x - y) \sin y$ is in the form of $\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)$.
Yes, and we know that $\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)=\sin(\alpha+\beta)$
So, $\sin (x - y) \cos y + \cos (x - y) \sin y=\sin((x-y)+y)$
$\sin((x-y)+y)=\sin(x)$