1. The range of the function $f(x)=\frac{\sin 2 x}{2+\sin x+\cos x}$ is
- 1. $\left[2 \sqrt{3}-4,1+\frac{\sqrt{2}}{2}\right]$.

Let $t=\sin x+\cos x$. Then $t=\sqrt{2} \sin \left(x+\frac{\pi}{4}\right) \in[-\sqrt{2}, \sqrt{2}]$.
Also, $\sin 2 x=t^{2}-1$, so, $f(x)=\frac{t^{2}-1}{2+t}$.
Let $m=t+2 \in[2-\sqrt{2}, 2+\sqrt{2}]$. Then $t=m-2$.
Thus, $f(x)=m+\frac{3}{m}-4$.
By the monotonicity of the double hook function, we have
$$
f(x) \in\left[2 \sqrt{3}-4,1+\frac{\sqrt{2}}{2}\right]
$$