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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?

mathreadler
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2 Answers2

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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.

Jack Crawford
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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.

Theo Bendit
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    Ah wow, the ancient Greeks of course. It is quite impossible to not be impressed how they managed to prove such things so early! Thank you. – mathreadler May 23 '19 at 11:34
  • There are geometrical proofs of trigonometric formulas, akin of those that could have been found by greek geometers, in particular addition formulas, in different places for example in https://math.stackexchange.com/q/443411, or in http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html – Jean Marie May 26 '19 at 08:36