Question

Evaluate: $5\left(\frac{{\cos }^2{60}^{\circ }+4{\sec }^2{30}^{\circ }-{\tan }^2{45}^{\circ }}{{\sin }^2{30}^{\circ }+{\cos }^2{30}^{\circ }}\right)$.

Answer 1

Given expression $5\left(\frac{{\cos }^2{60}^{\circ }+4{\sec }^2{30}^{\circ }-{\tan }^2{45}^{\circ }}{{\sin }^2{30}^{\circ }+{\cos }^2{30}^{\circ }}\right)$
$=5\left(\frac{(\frac{1}{2}{)}^2+4(\frac{2}{\sqrt{3}}{)}^2-(1{)}^2}{(\frac{1}{2}{)}^2+(\frac{\sqrt{3}}{2}{)}^2}\right)$
$=5\left(\frac{\frac{1}{4}+\frac{16}{3}-1}{\frac{1}{4}+\frac{3}{4}}\right)$
$=5\left(\frac{\frac{3+64-12}{12}}{\frac{1+3}{4}}\right)$
$=5\left(\frac{\frac{55}{12}}{1}\right)$
$=5\times \frac{55}{12}$
$=\frac{275}{12}$