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Evaluate each of the following:
\( \left(\operatorname{cosec}^{2} 45^{\circ} \sec ^{2} 30^{\circ}\right)\left(\sin ^{2} 30^{\circ}+4 \cot ^{2} 45^{\circ}-\sec ^{2} 60^{\circ}\right) \)
Given:
\( \left(\operatorname{cosec}^{2} 45^{\circ} \sec ^{2} 30^{\circ}\right)\left(\sin ^{2} 30^{\circ}+4 \cot ^{2} 45^{\circ}-\sec ^{2} 60^{\circ}\right) \)
To do:
We have to evaluate \( \left(\operatorname{cosec}^{2} 45^{\circ} \sec ^{2} 30^{\circ}\right)\left(\sin ^{2} 30^{\circ}+4 \cot ^{2} 45^{\circ}-\sec ^{2} 60^{\circ}\right) \).
Solution:
We know that,
$cosec 45^{\circ}=\sqrt2$
$\sec 30^{\circ}=\frac{2}{\sqrt3}$
$\sin 30^{\circ}=\frac{1}{2}$
$\cot 45^{\circ}=1$
$\sec 60^{\circ}=2$
$\left(\operatorname{cosec}^{2} 45^{\circ} \sec ^{2} 30^{\circ}\right)\left(\sin ^{2} 30^{\circ}+4 \cot ^{2} 45^{\circ}-\sec ^{2} 60^{\circ}\right)=\left[\left(\sqrt{2}\right)^{2}\left(\frac{2}{\sqrt{3}}\right)^{2}\right]\left[\left(\frac{1}{2}\right)^{2} +4( 1)^{2} -( 2)^{2}\right]$
$=\left( 2\times \frac{4}{3}\right)\left(\frac{1}{4} +4-4\right)$
$=\left(\frac{8}{3}\right)\left(\frac{1}{4}\right)$
$=\frac{2}{3}$
Hence, $\left(\operatorname{cosec}^{2} 45^{\circ} \sec ^{2} 30^{\circ}\right)\left(\sin ^{2} 30^{\circ}+4 \cot ^{2} 45^{\circ}-\sec ^{2} 60^{\circ}\right)=\frac{2}{3}$.