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Main Question or Discussion Point
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.

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 "proofread", or  in particular  offer proofs of the trig identities used, I would be grateful.
 "student"
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.

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 "proofread", or  in particular  offer proofs of the trig identities used, I would be grateful.
 "student"
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