Evaluate each of the following:$ \cos ^{2} 30^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} 90^{\circ} $

Given:

\( \cos ^{2} 30^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} 90^{\circ} \)

To do:

We have to evaluate \( \cos ^{2} 30^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} 90^{\circ} \).

Solution:  

We know that,

$\cos 30^{\circ}=\frac{\sqrt3}{2}$

$\cos 45^{\circ}=\frac{1}{\sqrt2}$

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

$\cos 90^{\circ}=0$

$ \cos ^{2} 30^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} 90^{\circ}=\left(\frac{\sqrt3}{2}\right)^{2} +\left(\frac{1}{\sqrt{2}}\right)^{2} +\left(\frac{1}{2}\right)^{2} +( 0)^{2}$

$=\frac{3}{4} +\frac{1}{2} +\frac{1}{4} $

$=\frac{3+1( 2) +1}{4}$

$=\frac{3+2+1}{4}$

$=\frac{6}{4}$

$=\frac{3}{2}$

Hence, $\cos ^{2} 30^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} 90^{\circ}=\frac{3}{2}$. 

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Updated on: 10-Oct-2022

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