Were the classic "high school" trigonometric formulae known and derived before Fourier transforms?

Question

We all probably learn the famous trigonometric formulae:

$$\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta) + \sin(\beta)\cos(\alpha)\\\cos(\alpha+\beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha) \sin(\beta)$$

Were these derived and widely known before the Fourier transform made it easy to prove them? If so, do we know who first proved them?

Answer 1

According to Wikipedia,

A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine.

Ptolemy lived about 100-170 AD, predating Fourier and his transform by a good 1500 years.

Answer 2

It is suggested on Wikipedia that Abu al-Wafa' Buzjani may have been the first to prove these identities around the 10th century AD, but apparently before then some Ancient Greek mathematicians had proved some equivalent identities in terms of chords. Trigonometry is very old and one of the earliest-studied disciplines.