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What is a non-geometric proof to prove the sine addition formula?

I know that method that using euler's constant or taylor's series works, but is there any others?

Search the google with the "non-geometric proof of sine addition formula" only provide me with the geometric way...

Anybody want to answer?

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How do you define the $\sin$ function ? – Belgi Oct 1 '12 at 22:05
@Belgi - The most basic one in High School. – Victor Oct 1 '12 at 22:06
If you are given a geomtric definition then the proof is also geometric. – Belgi Oct 1 '12 at 22:07
@Belgi - Any of the definition on – Victor Oct 1 '12 at 22:11
@Victor perhaps… will be of interest. In that thread we discuss definitions of sine... this sort of question always comes back to that. – James S. Cook Oct 2 '12 at 0:41
up vote 1 down vote accepted

Here is a non-geometric proof of DeMoivre's theorem that does not use Euler's theorem. But the result of section 3 (before DeMoivre's theorem is proved) is the angle addition formulas, if you equate components:

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