Question

Evaluate:

$\frac{5{\cos }^{2}60°+4{\sec }^{2}30°-{\tan }^{2}45°}{{\sin }^{2}30°+{\cos }^{2}30°}$

Answer 1

$$\begin{gathered} \frac{5{\cos }^{2}60°+4{\sec }^{2}30°-{\tan }^{2}45°}{{\sin }^{2}30°+{\cos }^{2}30°} \\ =\frac{5{\left({\displaystyle \frac{1}{2}}\right)}^2+4{\left({\displaystyle \frac{2}{\sqrt{3}}}\right)}^2-{\left(1\right)}^2}{{\left({\displaystyle \frac{1}{2}}\right)}^2+{\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^2} \\ =\frac{\left({\displaystyle \frac{5}{4}}+{\displaystyle \frac{4\times 4}{3}}-1\right)}{\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{3}{4}}\right)} \\ =\frac{\left({\displaystyle \frac{5}{4}+\frac{16}{3}-\frac{{\displaystyle 1}}{1}}\right)}{\left({\displaystyle \frac{4}{4}}\right)} \\ =\frac{\left({\displaystyle \frac{15+64-12}{12}}\right)}{\left({\displaystyle \frac{4}{4}}\right)} \\ =\frac{\left({\displaystyle \frac{67}{12}}\right)}{\left(1\right)} \\ =\frac{67}{12} \end{gathered}$$