$\begin{aligned} & \sqrt{3} \operatorname{cosec} x+2 \\ & \Rightarrow \sin x=-\frac{\sqrt{3}}{2}=-\sin \frac{\pi}{3} \\ & \Rightarrow x=\pi+\frac{\pi}{3} \text { or } 2 \pi-\frac{\pi}{3} \\ & \Rightarrow x=\frac{4 \pi}{3} \text { or } \frac{5 \pi}{3}\end{aligned}$