<s>[INST] Show that $\frac {\sin(3x)}{ \sin x} + \frac {\cos(3x)}{ \cos x} = 4\cos(2x)$ Show that 

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

 [/INST] $$\dfrac{\sin(3x)}{\sin x} + \frac{\cos(3x)}{\cos x}$$
$$\dfrac{\sin(3x)\cos x + \cos(3x)\sin x}{\cos x{\sin x}}$$
$$\dfrac {\sin (3x+x)}{\sin x\cos x}$$
$$\dfrac {\sin 4x}{\sin x\cos x}$$
$$\dfrac {2\sin 2x\cos 2x}{\sin x\cos x}$$
$$\dfrac {4\sin x\cos x\cos 2x}{\sin x\cos x}$$
$$4\cos 2x$$
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