Heavenly ax * If $\sin x+\sin y=\frac{\sqrt{2}}{2}, \cos x+\cos y=\frac{\sqrt{6}}{2}$, then $\sin (x+y)$ equals ( ).
(A) $\frac{\sqrt{2}}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\frac{\sqrt{6}}{2}$
(D) 1
Parse: Squaring the two expressions and adding them, we get
$$
\left(\sin ^{2} x+\cos ^{2} x\right)+\left(\sin ^{2} y+\cos ^{2} y\right)+2(\cos x \cdot \cos y+\sin x \cdot \sin y)=2,
$$

so $\cos (x-y)=0$. Multiplying the two expressions, we get
$$
(\sin x \cdot \cos y+\sin y \cdot \cos x)+(\sin x \cdot \cos x+\sin y \cdot \cos y)=\frac{\sqrt{3}}{2},
$$

which means $\sin (x+y)+\sin (x+y) \cdot \cos (x-y)=\frac{\sqrt{3}}{2}$. Therefore, $\sin (x+y)=\frac{\sqrt{3}}{2}$, hence the answer is B.