<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] We have,
$$\sin(3x) = 3\sin(x) - 4\sin^{3}(x)$$
and
$$\cos(3x) = 4\cos^{3}(x) - 3 \cos(x).$$
Thus, your expression reduces to 
$$3 - 4\sin^{2}(x) + 4\cos^{2}(x) - 3$$
which is equal to
$$4 \cos(2x),$$
as $\cos(2x) = \cos^{2}(x) - \sin^{2}(x)$.
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