Find the values of

(i) $\dfrac{\text{sin}^2 45° + \text{cos}^2 45°}{\text{tan}^2 60°}$

(ii) $\dfrac{\text{sin 30° - sin 90° + 2 cos 0°}}{\text{tan 30° × tan 60°}}$

(iii) $\dfrac{4}{3} \text{tan}^2 30° + \text{sin}^2 60° - \text{3 cos}^2 60° + \dfrac{3}{4}\text{tan}^2 60° - 2\text{tan}^2 45°$.

(i) Solving,

$\Rightarrow \dfrac{\text{sin}^2 45° + \text{cos}^2 45°}{\text{tan}^2 60°} \\[1em] [\because \text{sin}^2 \text{ θ} + \text{cos}^2 \text{ θ} = 1], \\[1em] \Rightarrow \dfrac{1}{(\sqrt{3})^2} \\[1em] \Rightarrow \dfrac{1}{3}$

Hence, $\dfrac{\text{sin}^2 45° + \text{cos}^2 45°}{\text{tan}^2 60°} = \dfrac{1}{3}$.

(ii) Solving,

$\Rightarrow \dfrac{\text{sin 30° - sin 90° + 2 cos 0°}}{\text{tan 30° × tan 60°}} \\[1em] \Rightarrow \dfrac{\dfrac{1}{2} - 1 + 2 \times 1}{\dfrac{1}{\sqrt{3}} \times \sqrt{3}} \\[1em] \Rightarrow \dfrac{\dfrac{1 - 2 + 4}{2}}{1} \\[1em] \Rightarrow \dfrac{3}{2}.$

Hence, $\dfrac{\text{sin 30° - sin 90° + 2 cos 0°}}{\text{tan 30° × tan 60°}} = \dfrac{3}{2}$.

(iii) Solving,

$\Rightarrow\dfrac{4}{3} \text{tan}^2 30° + \text{sin}^2 60° - \text{3 cos}^2 60° + \dfrac{3}{4}\text{tan}^2 60° - 2\text{tan}^2 45° \\[1em] \Rightarrow \dfrac{4}{3} \times \Big(\dfrac{1}{\sqrt{3}}\Big)^2 + \Big(\dfrac{\sqrt{3}}{2}\Big)^2 - 3 \times \Big(\dfrac{1}{2}\Big)^2 + \dfrac{3}{4} \times (\sqrt{3})^2 - 2 \times (1)^2 \\[1em] \Rightarrow \dfrac{4}{3} \times \dfrac{1}{3} + \dfrac{3}{4} - \dfrac{3}{4} + \dfrac{3}{4} \times 3 - 2 \\[1em] \Rightarrow \dfrac{4}{9} + \dfrac{9}{4} - 2 \\[1em] \Rightarrow \dfrac{16 + 81 - 72}{36} \\[1em] \Rightarrow \dfrac{25}{36}.$

Hence, $\dfrac{4}{3} \text{tan}^2 30° + \text{sin}^2 60° - \text{3 cos}^2 60° + \dfrac{3}{4}\text{tan}^2 60° - 2\text{tan}^2 45° = \dfrac{25}{36}$.