Evaluate each of the following

We have,

$\operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ}$....(1)

Now,

$\operatorname{cosec} 30^{\circ}=2, \cos 60^{\circ}=\frac{1}{2}, \sec 45^{\circ}=\sqrt{2}, \tan 45^{\circ}=1, \sin 90^{\circ}=1, \cot 30^{\circ}=\sqrt{3}$

So by substituting above values in equation (1)

We get,

$\operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ}$

$=(2)^{3} \times\left(\frac{1}{2}\right) \times(1)^{3} \times(1)^{2} \times(\sqrt{2})^{2} \times(\sqrt{3})$

$=8 \times\left(\frac{1}{2}\right) \times 1 \times 1 \times 2 \times(\sqrt{3})$

$=\frac{8}{2} \times 2 \times(\sqrt{3})$

Now, 2 gets cancelled and we get,

$\operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ}=8 \sqrt{3}$