Evaluate each of the following

Solution:

We have to find the following expression

$\cos ^{2} 30^{\circ}+\cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} 90^{\circ} \ldots \ldots$ (1)

Now,

$\cos 45^{\circ}=\frac{1}{\sqrt{2}}, \cos 30^{\circ}=\frac{\sqrt{3}}{2}, \cos 60^{\circ}=\frac{1}{2}, \cos 90^{\circ}=0$

So by substituting above values in equation (1)

We get,

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

$=\left(\frac{\sqrt{3}}{2}\right)^{2}+\left(\frac{1}{\sqrt{2}}\right)^{2}+\left(\frac{1}{2}\right)^{2}+(0)^{2}$

$=\frac{(\sqrt{3})^{2}}{2^{2}}+\frac{1^{2}}{(\sqrt{2})^{2}}+\frac{1^{2}}{2^{2}}+0$

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

Now by taking denominator 4 together and simplifying

We get,

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

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

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

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

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

Now by taking LCM

We get,

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

$=\frac{1 \times 2}{1 \times 2}+\frac{1}{2}$

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

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

$=\frac{3}{2}$

Therefore,

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