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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,
$cosec 30^{\circ}=2$
$\cos 60^{\circ}=\frac{1}{2}$
$\tan 45^{\circ}=1$
$\sin 90^{\circ}=1$
$\sec 45^{\circ}=\sqrt2$
$\cot 30^{\circ}=\sqrt3$
Therefore,$\operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ}=( 2)^{3} \times \left(\frac{1}{2}\right) \times ( 1)^{3} \times ( 1)^{2} \times \left(\sqrt{2}\right)^{2} \times \left(\sqrt{3}\right)$
$=8\times \left(\frac{1}{2}\right) \times ( 1) \times ( 1) \times 2\times \sqrt{3}$
$=8\sqrt{3}$
Hence, $\operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ}=8\sqrt3$.