Evaluate the following :

$\dfrac{\text{cos 45°}}{\text{sec 30° + cosec 30°}}$

Substituting values, we get :

$\Rightarrow \dfrac{\text{cos 45°}}{\text{sec 30° + cosec 30°}} = \dfrac{\dfrac{1}{\sqrt{2}}}{\dfrac{2}{\sqrt{3}} + 2} \\[1em] = \dfrac{\dfrac{1}{\sqrt{2}}}{\dfrac{2 + 2\sqrt{3}}{\sqrt{3}}} \\[1em] = \dfrac{\sqrt{3}}{\sqrt{2}(2 + 2\sqrt{3})} \\[1em] = \dfrac{\sqrt{3}}{2\sqrt{2}(1 + \sqrt{3})} \\[1em] = \dfrac{\sqrt{3}}{2\sqrt{2}(1 + \sqrt{3})} \times \dfrac{1 - \sqrt{3}}{1 - \sqrt{3}} \\[1em] = \dfrac{\sqrt{3}(1 - \sqrt{3})}{2\sqrt{2}(1 - 3)} \\[1em] = \dfrac{\sqrt{3}(1 - \sqrt{3})}{2\sqrt{2} \times -2} \\[1em] = \dfrac{-\sqrt{3}(1 - \sqrt{3})}{4\sqrt{2}} \\[1em] = \dfrac{3 - \sqrt{3}}{4\sqrt{2}} \\[1em] = \dfrac{3 - \sqrt{3}}{4\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} \\[1em] = \dfrac{3\sqrt{2} - \sqrt{6}}{8}.$

Hence, $\dfrac{\text{cos 45°}}{\text{sec 30° + cosec 30°}} = \dfrac{3\sqrt{2} - \sqrt{6}}{8}.$