Natural applications of the Sum-to-Product formulas

Question

The Sum-to-Product formulas are formulas used to express sums and differences of $\sin$ and $\cos$, for example

$$\cos(p)+\cos(q)=2\cos(\frac{p+q}{2})\cos(\frac{p-q}{2})$$

Are there any good, motivational application of these formulas? All applications I found seem to be quite artificial, like:

1. solve the equation $\sin(5x)+\sin(3x)=0$
2. Compute $\cos(195^o)+\cos(105^o)$.

Since the Sum-to-Product formulas can be proved from the Sum and Difference formulas $\cos(a+b)$ and al., I can solve the second problem by computing $\cos(195^o)$ and $\cos(105^o)$. That's a little bit longer but I would have the satisfaction of knowing the cosine of two new angles, rather than the result of a sum that has been made on purpose. The first one is equivalent to $\sin(3x)=-\sin(5x)$, that one can easily solve without knowing the tricky (is it a $\cos$? a $\sin$? of $\frac{p+q}{2}$ or of $\frac{p-q}{2}$?) Sum-to-Product formulas.

Why should I learn the Sum-to-Product formulas?

PS: before voting to close because of "too broad" or "asking for a list", please try for a few minutes to find applications or problems that don't seem artificial. You will see that there are not many of them.

Answer 1

As I have suggested in a comment, don't memorise these formulae.

However, a suggestion as to why they are useful:

for easily integrating functions of the form $\cos(ax)\cos(bx)$ etc;

which are important because

they occur in Fourier series;

which are important because

they are needed to solve partial differential equations involving heat, vibrating strings, vibrating membranes and many many other things.