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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 ?2012-10-01
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    @Belgi - The most basic one in High School.2012-10-01
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    If you are given a geomtric definition then the proof is also geometric.2012-10-01
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    @Belgi - Any of the definition on http://mathworld.wolfram.com/Sine.html2012-10-01
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    Did you study calculus before ? some of the definitions are with power series2012-10-01
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    @Belgi - Except the euler's constant and taylor's series method...2012-10-01
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    @Victor perhaps http://math.stackexchange.com/questions/189016/deriving-the-addition-formula-of-sin-u-from-a-total-differential-equation/189101#189101 will be of interest. In that thread we discuss definitions of sine... this sort of question always comes back to that.2012-10-02
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    @JamesS.Cook - Any other way?2012-10-02

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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: http://www.dfcd.net/articles/demoivre.pdf