Talk:Cosine of Sum

The Geometric Proof is the only valid proof.

Proof by Algebra and Proof by Euler's Formula are both invalid proofs.

The reason I say this is because, if you truly respect the notion of rigorous proof, these two proofs I criticize are literally circular in logic.

In order to prove Euler's Formula or the Maclaurin Series for sine and cosine, you must know how to take the derivatives of sine and cosine. There is simply no other way to prove Euler's Formula or the Maclaurin series except en route of the derivatives of sine and cosine.

Furthermore, how do you prove the derivatives of sine and cosine? Via the limit definition of a derivative, one must already know how to find the sine and cosine of a sum and a difference.

Thus, what we have here is a case in which we prove one identity with a second and a second identity with the first.

This proof page uses consequence of truth as a premise to prove. The premises of these proofs are "corollaries", so to speak, if you delve deep enough into math, far beyond the level of math in which these trigonometric identities would be proved.

Don't get me wrong, these so-called "proofs" are marvelous demonstrations... but a demonstration is all that it is worth.

I wonder how many other Proof Wiki pages are logically corrupted?!?!

--CogitoErgoCogitoSum 20:34, 8 December 2009 (UTC)

That's a bit dogmatic. Have you traced through the chain of proofs? For the "proof by algebra", the sine and cosine are here defined by the series given. They are not derived from Euler's Formula or Maclaurin series. They are defined as they are, and the properties of sine and cosine are deduced directly from those definitions.

As an approach it's as valid as the geometrical approach, but leads itself more easily to an axiomatic approach from which you start with ZFC, define the natural numbers (still work in progress), then integers and so on up to reals, then derive the whole of analysis from understanding basic properties of the real numbers.

From that point of view, the reasoning is not circular.

Alternatively, you can start with the geometrical proofs, as you suggest, but to do this one needs to get the basics of geometry out of the way (and we're barely started on Euclid yet, haven't finished his first book, we're lazy so-and-so's on this site, we get bored easily).

Or you can start with Euler's Identity, which can be derived from a different axiomatic framework (but as I didn't put that proof up, I can't vouch for the details of what direction this will come from).--Matt Westwood 22:40, 8 December 2009 (UTC)