Question

Evaluate each of the following :
$4(si{n}^4{30}^{\circ }+co{s}^2{60}^{\circ })-3(co{s}^2{45}^{\circ }-si{n}^2{90}^{\circ })-si{n}^2{60}^{\circ }$

Answer 1

$4(si{n}^4{30}^{\circ }+co{s}^2{60}^{\circ })-3(co{s}^2{45}^{\circ }-si{n}^2{90}^{\circ })-si{n}^2{60}^{\circ }$

$=4({\left(\frac{1}{2}\right)}^4+{\left(\frac{1}{2}\right)}^2)-3({\left(\frac{1}{\sqrt{2}}\right)}^2-1^2)-{\left(\frac{\sqrt{3}}{2}\right)}^2$

$=4(\left(\frac{1}{16}\right)+\left(\frac{1}{4}\right))-3(\left(\frac{1}{2}\right)-1)-\left(\frac{3}{4}\right)$

$=4(\left(\frac{1}{16}\right)+\left(\frac{4}{16}\right))-3(\left(\frac{1}{2}\right)-1)-\left(\frac{3}{4}\right)$

$=4\left(\frac{5}{16}\right)-3\left(\frac{-1}{2}\right)-\left(\frac{3}{4}\right)$

$=\frac{5}{4}+\frac{3}{2}-\frac{3}{4}$

$=\frac{5}{4}-\frac{3}{4}+\frac{3}{2}$

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

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

$=\frac{4}{2}$

$=2$