Simplify: $$\frac{\sin 3A}{\sin A} + \frac{\cos 3A}{\cos A}$$​​

Correct option is D

Given:

$$\frac{\sin 3A}{\sin A} + \frac{\cos 3A}{\cos A}$$   

Formula Used:

$$\sin(a + b) = \sin A \cdot \cos B + \cos A \cdot \sin B\\\sin 2A = 2 \sin A \cdot \cos A$$   

Solution:

$$\frac{\sin 3A}{\sin A} + \frac{\cos 3A}{\cos A}\\\ \\= \frac{(\sin 3A \cdot \cos A + \cos 3A \cdot \sin A)}{\sin A \cdot \cos A}\\\ \\= \frac{\sin (3A + A)}{\sin A \cdot \cos A}\\\ \\= \frac{2\sin 4A}{2\sin A \cdot \cos A}\\\ \\= \frac{2 \cdot 2\sin 2A \cdot \cos 2A}{\sin 2A}\\\ \\= 4\cos 2A$$​