Are you objecting to the trivial extension to cos(α - β)? The cos(α + β) identity itself is nicely explained in the text.

The third thing you quote is the result of fairly simply symbol manipulation that requires no new knowledge apart from the original cosine identity and the obvious corollary about subtracting an angle being the same as adding a negated one.

There is zero advanced math there. No complex numbers, no calculus, no limits, no Fourier, no "functions are vectors, too".