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 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

By, means of an example, show that sin(A + B) ≠ sin A + sin B.

If A = 60° and B = 30°, verify that

(i) sin(A + B) = sin A cos B + cos A sin B

(ii) cos(A + B) = cos A cos B - sin A sin B

(iii) sin(A - B) = sin A cos B - cos A sin B

(iv) tan(A - B) = $\dfrac{\text{tan A - tan B}}{\text{1 + tan A tan B}}.$