Sums to Products

Read the product formulas backwards, with the half-sum and the half-difference of the angles, and a sum becomes a product:

$$\sin\left(A\right)+\sin\left(B\right) = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$$

$$\cos\left(A\right)+\cos\left(B\right) = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$$

For $\sin\left(5x\right)+\sin\left(x\right)$, the half-sum is 3x and the half-difference is 2x, so it is $2\sin\left(3x\right)\cos\left(2x\right)$.

This is how two notes of nearly the same pitch sound together. Their sum is a fast wave at the average pitch times a slow wave at half the difference in pitch. The loudness follows the size of the slow wave, which peaks twice in each of its cycles, so the beat you hear repeats at the full difference: notes of 440 Hz and 442 Hz beat twice a second.