Question

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

Answer 1

Let,

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

$={\left(\frac{\sqrt{3}}{2}\right)}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2+0^2$ $[\because \cos {45}^{\circ }=\frac{1}{\sqrt{2}},\cos {30}^{\circ }=\frac{\sqrt{3}}{2},\cos {60}^{\circ }=\frac{1}{2},\cos {90}^{\circ }=0]$

$=\frac{3}{4}+\frac{1}{2}+\frac{1}{4}+0$

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

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

$=\frac{6}{4}$

$=\frac{3}{2}$