Find the values of :

$\dfrac{\text{2 cos}^2 45° + \text{3 tan}^2 30°}{\sqrt{3}\text{cos 30° + \text{sin 30°}}}$

Solving,

$\Rightarrow \dfrac{\text{2 cos}^2 45° + \text{3 tan}^2 30°}{\sqrt{3}\text{cos 30° + \text{sin 30°}}} \\[1em] \Rightarrow \dfrac{2 \times \Big(\dfrac{1}{\sqrt{2}}\Big)^2 + 3 \times \Big(\dfrac{1}{\sqrt{3}}\Big)^2}{\sqrt{3} \times \dfrac{\sqrt{3}}{2} + \dfrac{1}{2}} \\[1em] \Rightarrow \dfrac{2 \times \dfrac{1}{2} + 3 \times \dfrac{1}{3}}{\dfrac{3}{2} + \dfrac{1}{2}} \\[1em] \Rightarrow \dfrac{1 + 1}{\dfrac{4}{2}} \\[1em] \Rightarrow 2 \times \dfrac{2}{4} \\[1em] \Rightarrow \dfrac{4}{4} \\[1em] \Rightarrow 1.$

Hence, $\dfrac{\text{2 cos}^2 45° + \text{3 tan}^2 30°}{\sqrt{3}\text{cos 30° + \text{sin 30°}}}$ = 1.