1. Let $3 \sin y=2 \sin ^{2} y+\sin ^{2} x$, then the range of $\cos 2 x-2 \sin y$ is
1. $[-2,1] \cup\{-3\}$ Detailed explanation: From $0 \leqslant \sin ^{2} x=-2 \sin ^{2} y+3 \sin y \leqslant 1$, we get $0 \leqslant \sin y \leqslant \frac{1}{2}$ or $\sin y=1$, thus $\cos 2 x-2 \sin y=1-2 \sin ^{2} x-2 \sin y=1$ $2\left(-2 \sin ^{2} y+3 \sin y\right)-2 \sin y=4 \sin ^{2} y-8 \sin y+1=4(\sin y-1)^{2}-3$, hence the range of $\cos 2 x-2 \sin y$ is $[-2,1] \cup\{-3\}$.