Evaluate the following :

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

Solving,

$\Rightarrow \dfrac{5\text{ cos}^2 60° + 4\text{ sec}^2 30° - \text{ tan}^2 45°}{\text{sin}^2 30° + \text{ cos}^2 30°} \\[1em] \Rightarrow \dfrac{5 \times \Big(\dfrac{1}{2}\Big)^2 + 4 \times \Big(\dfrac{2}{\sqrt{3}}\Big)^2 - 1^2}{\Big(\dfrac{1}{2}\Big)^2 + \Big(\dfrac{\sqrt{3}}{2}\Big)^2} \\[1em] \Rightarrow \dfrac{5 \times \dfrac{1}{4} + 4 \times \dfrac{4}{3} - 1}{\dfrac{1}{4} + \dfrac{3}{4}} \\[1em] \Rightarrow \dfrac{\dfrac{5}{4} + \dfrac{16}{3} - 1}{\dfrac{4}{4}} \\[1em] \Rightarrow \dfrac{\dfrac{15 + 64 - 12}{12}}{1} \\[1em] \Rightarrow \dfrac{67}{12}.$

Hence, $\dfrac{5\text{ cos}^2 60° + 4\text{ sec}^2 30° - \text{ tan}^2 45°}{\text{sin}^2 30° + \text{ cos}^2 30°} = \dfrac{67}{12}$.