<s>[INST] How does $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ become $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$? How can you rewrite $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ to $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$?
What rules have been used? I need it on this form to compare it with a fourier series.
 [/INST] Hint. One may use
$$\begin{align}
\cos a \cos b &=\frac12\left((\cos (a-b)+\cos (a+b)\right).
\end{align}$$
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