Problem Statement
Prove that for all values of $\theta$ where $\sin\theta \ne 0$ and $\cos\theta \ne 0$:
$$\frac{\sin 3\theta}{\sin\theta} - \frac{\cos 3\theta}{\cos\theta} = 2$$ [5 Marks]
Verified Solution & Marking Scheme
Combine fractions over common denominator
$$\text{LHS} = \frac{\sin 3\theta\cos\theta - \cos 3\theta\sin\theta}{\sin\theta\cos\theta}$$
Apply Compound Angle Identity in Numerator
Recall $\sin(A - B) = \sin A\cos B - \cos A\sin B$. Setting $A = 3\theta$ and $B = \theta$:
$$\sin 3\theta\cos\theta - \cos 3\theta\sin\theta = \sin(3\theta - \theta) = \sin(2\theta)$$
Apply Double Angle Identity & Simplify
$$\text{LHS} = \frac{\sin 2\theta}{\sin\theta\cos\theta} = \frac{2\sin\theta\cos\theta}{\sin\theta\cos\theta} = 2 = \text{RHS} \quad \text{(Q.E.D.)}$$