Show that $\frac {\sin(3x)}{ \sin x} + \frac {\cos(3x)}{ \cos x} = 4\cos(2x)$ Show that 

$$\frac{\sin(3x)}{\sin x} + \frac{\cos(3x)}{\cos x} = 4\cos(2x).$$



$$\frac {\sin3x} {\sin x} = 3-4\sin^2x$$

as $\sin3x = 3\sin x-4\sin^3x$

$$\frac {\cos3x} {\cos x} = 4\cos^2x - 3$$

as $\cos3x = 4\cos^3x -3\cos x$
adding the two above we get $4(\cos^2x - \sin^2x)=4\cos2x.$
