Analytic proof of continuity, differentiability of trig. functions

[student10567]

Since I am new to PF (hi!), before I go any further, I would like to

a) briefly note that this is an independent study question, and that its scope goes beyond that of a textbook question - i.e., I believe that this thread belongs here - and
b) also note that I am new to analysis and early in my calculus education, so I do not have an understanding of infinite series which is sufficiently developed for me to define trigonometric functions as series. I know that this precludes some of the easier options for doing what I am trying to.

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I am interested in demonstrating (analytically, as much as possible) the continuity and differentiability of the trigonometric functions.

To that end, I completed a proof of these results myself (attachment); but some elements of this proof make me uncomfortable.

Among these are:

- I had to cite Euclidean geometry as the basis of my proof.
- I don't have rigorous proofs for many identities (including oddness/evenness of the functions).
- I feel unsure about my generalization of my work (p. 6 or so) from a closed range (0, 2∏) to ℝ.

If anyone with a deeper understanding of the trigonometric functions could help "proof-read", or - in particular - offer proofs of the trig identities used, I would be grateful.

- "student"

 

[resolvent1]

Well, if you're interested in doing them analytically as mich as possible, then the natural way to solve it is to use the Taylor series expansions (preferably centered at 0) for the trig functions. (naturally because these functions are analytic).

 

[student10567]

I would like to use the Taylor series, but this method is essentially unavailable to me right now (note (b) at the top).