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, $\left(\operatorname{cosec}^{2} 45^{\circ} \sec ^{2} 30^{\circ}\right)\left(\sin ^{2} 30^{\circ}+4 \cot ^{2} 45^{\circ}-\sec ^{2} 60^{\circ}\right)=\frac{2}{3}$. 

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

26 Views

Kickstart Your Career

Get certified by completing the course

Get Started