If $\sin x+\sin y=a$ and $\cos x+\cos y=b$, then find the value of $(\sin x\sin y+\cos x\cos y)$.

Given the equations:
1. $\sin x+\sin y=a$
2. $\cos x+\cos y=b$

Squaring equation (1) on both sides:

$${\sin }^{2}x+{\sin }^{2}y+2\sin x\sin y=a^2$$

Squaring equation (2) on both sides:

$${\cos }^{2}x+{\cos }^{2}y+2\cos x\cos y=b^2$$

Adding both squared equations:

$$({\sin }^{2}x+{\cos }^{2}x)+({\sin }^{2}y+{\cos }^{2}y)+2(\sin x\sin y+\cos x\cos y)=a^2+b^2$$

Using the fundamental identity ${\sin }^2\theta +{\cos }^2\theta =1$:

$$1+1+2(\sin x\sin y+\cos x\cos y)=a^2+b^2$$

$$2+2(\sin x\sin y+\cos x\cos y)=a^2+b^2$$

$$2(\sin x\sin y+\cos x\cos y)=a^2+b^2-2$$

$$\sin x\sin y+\cos x\cos y=\frac{a^2+b^2-2}{2}$$