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With the Taylor series representation of $\sin$ or $\cos$ as a starting point (and assuming no other knowledge about those functions), how can one:

a. prove they are periodic?

b. find the value of the period?

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    The proof's somewhere in chapter 2 of Ahlfors's *Complex Analysis*, but I don't have my copy with me to check.2011-09-09
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    [Quite related...](http://math.stackexchange.com/questions/33732)2011-09-09
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    English aside: **you don't know nothing else** ... You mean **we know nothing else** or **we don't know anything else** .2011-09-09
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    @Pierre: GEdgar is correct that language purists will sneer at your double negative. But double negatives are a regular feature of many dialects of English, so don't be downhearted. GEdgar is wrong, however, in insisting on the first person plural ('we'). The second person ('you') is preferable here, because in fact we *do* know more about these functions.2011-09-09
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    @Tony: in that case, "and without assuming anything else" might be a better choice of words, no?2011-09-09
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    @Tony, linguistic peevery about double negatives is one thing -- but in mathematics we need to place greater demands on precision than ordinary language does. This is if for the pragmatic reason that we sometimes do have to utter a negated negation and have it understood as such. And, also in contrast to most nonmathematical conversation, we cannot rely on common sense to disambiguate, because we sometimes _deliberately utter falsehoods_ for the purpose of proving things about these falsehoods.2011-09-09
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    I think a proof of this is in Baby Rudin (i.e. Walter Rudin's _Principles of Mathematical Analysis_). Maybe in the "special functions" chapter.2011-09-09
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    @J.M.: Yes, your version is probably best. But OP's choice of pronoun was better than GEdgar's.2011-09-09
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    This is done in the Prologue to Rudin _Real and Complex Analysis_. You can find a more detailed exposition of the same argument in the appendix on exp, sin and cos in _Complex Made Simple_.2018-01-05

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