Example 1 Given $\left\{\begin{array}{l}\sin \alpha+\sin \beta=1 \\ \cos \alpha+\cos \beta=0\end{array}\right.$, find $\cos 2 \alpha+\cos 2 \beta$.
Solution: From the given, we have $\left\{\begin{array}{l}\sin \alpha=1-\sin \beta, \\ \cos \alpha=-\cos \beta \text {. }\end{array}\right.$ Squaring and adding, we get $\sin \beta=\frac{1}{2}$, thus obtaining $\sin \alpha=\frac{1}{2}$.
Therefore, $\cos 2 \alpha+\cos 2 \beta=1-2 \sin ^{2} \alpha+1-2 \sin ^{2} \beta=1$.