Evaluate each of the following:$ \operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ} $

Given:

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

To do:

We have to evaluate \( \operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ} \).

Solution:  

We know that,

$cosec 30^{\circ}=2$

$\cos 60^{\circ}=\frac{1}{2}$

$\tan 45^{\circ}=1$

$\sin 90^{\circ}=1$

$\sec 45^{\circ}=\sqrt2$

$\cot 30^{\circ}=\sqrt3$

$\operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ}=( 2)^{3} \times \left(\frac{1}{2}\right) \times ( 1)^{3} \times ( 1)^{2} \times \left(\sqrt{2}\right)^{2} \times \left(\sqrt{3}\right)$

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

$=8\sqrt{3}$

Hence, $\operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ}=8\sqrt3$.  

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Updated on: 10-Oct-2022

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