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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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