$\begin{aligned} & \sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ} \\ &=\frac{\tan 60^{\circ}}{\sin 20^{\circ}}-\frac{1}{\cos 20^{\circ}} \\ &=\frac{\sin 60^{\circ} \cos 20^{\circ}-\sin 20^{\circ} \cos 60^{\circ}}{\cos 60^{\circ} \sin 20^{\circ} \cos 20^{\circ}} \\ &=\frac{\sin 40^{\circ}}{\cos 60^{\circ} \sin 20^{\circ} \cos 20^{\circ}} \\ &=\frac{2 \sin 20^{\circ} \cos 20^{\circ}}{\frac{1}{2}\left(\sin 20^{\circ} \cos 20^{\circ}\right)}=4\end{aligned}$