tan 30° cosec 60° + tan 60° sec 30°

As we know that,

$\tan 30^{\circ}=\frac{1}{\sqrt{3}}$

$\sec 30^{\circ}=\frac{2}{\sqrt{3}}$

$\operatorname{cosec} 60^{\circ}=\frac{2}{\sqrt{3}}$

$\tan 60^{\circ}=\sqrt{3}$

By substituting these values, we get

$\tan 30^{\circ} \operatorname{cosec} 60^{\circ}+\tan 60^{\circ} \sec 30^{\circ}=\frac{1}{\sqrt{3}} \times \frac{2}{\sqrt{3}}+\sqrt{3} \times \frac{2}{\sqrt{3}}$

$=\frac{2}{3}+2$

$=\frac{2+6}{3}$

$=\frac{8}{3}$

Hence, $\tan 30^{\circ} \operatorname{cosec} 60^{\circ}+\tan 60^{\circ} \sec 30^{\circ}=\frac{8}{3}$