& \sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ}=\frac{\sqrt{3}}{\sin 20^{\circ}}-\frac{1}{\cos 20^{\circ}} \\ &=\frac{\sqrt{3} \cos 20^{\circ}-\sin 20^{\circ}}{\sin 20^{\circ} \cos 20^{\circ}} \\ &=\frac{2\left(\frac{\sqrt{3}}{2} \cos 20^{\circ}-\frac{1}{2} \sin 20^{\circ}\right)}{\sin 20^{\circ} \cos 20^{\circ}} \end{aligned}$
$=\frac{2\left(\cos 30^{\circ} \cos 20^{\circ}-\sin 30^{\circ} \sin 20^{\circ}\right)}{\left(\frac{2 \sin 20^{\circ} \cos 20^{\circ}}{2}\right)}$
$=\frac{4\left[\cos \left(30^{\circ}+20^{\circ}\right)\right]}{\sin 40^{\circ}}$
$=\frac{4 \cos 50^{\circ}}{\sin 40^{\circ}}=\frac{4 \cos \left(90^{\circ}-40^{\circ}\right)}{\sin 40^{\circ}}=\frac{4 \sin 40^{\circ}}{\sin 40^{\circ}}=4 .$