Prove that

(i) cos² 30° + sin 30° + tan² 45° = $2\dfrac{1}{4}$

(ii) 4(sin⁴ 30° + cos⁴ 60°) - 3(cos² 45° - sin² 90°) = 2

(iii) cos 60° = cos² 30° - sin² 30°.

If θ = 30°, verify that

(i) sin 2θ = 2 sin θ cos θ

(ii) cos 2θ = 2 cos² θ - 1

(iii) sin 3θ = 3 sin θ - 4 sin³ θ

(iv) cos 3θ = 4 cos³ θ - 3 cos θ

Find the values of

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

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

If x = 30°, verify that tan 2x = $\dfrac{\text{2 tan x}}{\text{1 - tan}^2 x}$.