Question

Evaluate each of the following :
$\frac{{\tan }^2{60}^{\circ }+4{\cos }^2{45}^{\circ }+3{\sec }^2{30}^{\circ }+5{\cos }^2{90}^{\circ }}{\text{cosec}{30}^{\circ }+\sec {60}^{\circ }-{\cot }^2{30}^{\circ }}$

Answer 1

Given

$\frac{{\tan }^2{60}^{\circ }+4{\cos }^2{45}^{\circ }+3{\sec }^2{30}^{\circ }+5{\cos }^2{90}^{\circ }}{\csc {30}^{\circ }+\sec {60}^{\circ }-{\cot }^2{30}^{\circ }}$

$=\frac{{\left(\sqrt{3}\right)}^2+4{\left(\frac{1}{\sqrt{2}}\right)}^2+3{\left(\frac{2}{\sqrt{3}}\right)}^2+5\left(0\right)}{2+2-{\left(\sqrt{3}\right)}^2}$

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

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

$=3+2+4$

$=9$