Find the values of

(i) $\sqrt{\dfrac{1 - \text{cos}^2 30°}{1 - \text{sin}^2 30°}}$

(ii) $\dfrac{\text{sin 45° cos 45° cos 60°}}{\text{sin 60° cos 30° tan 45°}}$

(i) Solving,

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

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

(ii) Solving,

$\Rightarrow \dfrac{\text{sin 45° cos 45° cos 60°}}{\text{sin 60° cos 30° tan 45°}} \\[1em] \Rightarrow \dfrac{\dfrac{1}{\sqrt{2}} \times \dfrac{1}{\sqrt{2}} \times \dfrac{1}{2}}{\dfrac{\sqrt{3}}{2} \times \dfrac{\sqrt{3}}{2} \times 1} \\[1em] \Rightarrow \dfrac{\dfrac{1}{4}}{\dfrac{3}{4}} \\[1em] \Rightarrow \dfrac{1}{3}.$

Hence, $\dfrac{\text{sin 45° cos 45° cos 60°}}{\text{sin 60° cos 30° tan 45°}} = \dfrac{1}{3}.$