Question

Evaluate:
$\frac{5{\sin }^2{30}^{\circ }+{\cos }^2{45}^{\circ }-4{\tan }^2{30}^{\circ }}{2\sin {30}^{\circ }\cos {30}^{\circ }+\tan {45}^{\circ }}$

Answer 1

We have,

$\frac{5si{n}^2{30}^{\circ }+co{s}^2{45}^{\circ }-4ta{n}^2{30}^{\circ }}{2sin{30}^{\circ }cos{30}^{\circ }+tan{45}^{\circ }}$

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

$=\frac{\frac{5}{4}+\frac{1}{2}-\frac{4}{3}}{\frac{\sqrt{3}}{2}+1}=\frac{\frac{5}{12}}{\frac{\sqrt{3}+2}{2}}=\frac{5}{12}\times \frac{2}{(\sqrt{3}+2)}=\frac{5}{6(2+\sqrt{3})}$