Why so Series

Notational Notions has an entry raising to the challenge of showing with power series that (sin x)² + (cos x)² = 1. I jestingly but unfairly called the author a cheater for using what to me was an obvious appeal to Euler's formula (I made a mistake in naming it as De Moivre's). I reasoned thus:

  (sin x)² + (cos x)²
= { Euler: e^ix - e^(-ix) = 2i sin x }
  [(e^ix - e^(-ix))/(2i)]² + [(e^ix + e^(-ix))/2]²
= { Binomial, twice }
 -(e^2ix - 2 e^ix e^(-ix) + e^(-2ix))/4 + (e^2ix + 2 e^ix e^(-ix) + e^(-2ix))/4
= { Algebra }
  (-e^2ix + 2 e^ix e^(-ix) - e^(-2ix) + e^2ix + 2 e^ix e^(-ix) + e^(-2ix))/4
= { Algebra throughout }
  (2 e^ix e^(-ix) + 2 e^ix e^(-ix))/4
= { Algebra }
  e^ix e^(-ix)
= { Negative power is inverse }
  1

The author defended the choice of purely imaginary exponentials as the most natural representation for the real series (1, 0, -1, 0, …). Meeting the challenge of showing a purely real derivation incurred in an ugly and possibly unavoidable case analysis. I conceded, but tried my hand at it. It is well-known that the series development for the sine and cosine are:

sin x = x - x^3/3! + x^5/5! - x^7/7! + …

cos x = 1 - x^2/2! + x^4/4! - x^6/6! + …

Squaring the first series:

 (sin x)²
= { Definition }
  (x - x^3/3! + x^5/5! - x^7/7! + …)²
= { Distribution }
      x^2/1!1! - x^4/3!1! + x^6/5!1! - x^8/7!1! + …
               - x^4/1!3! + x^6/3!3! - x^8/5!3! + x^10/3!7! - …
                          + x^6/1!5! - x^8/3!5! + x^10/5!5! - x^12/7!5! + …
                                     - x^8/1!7! + x^10/7!3! - x^12/7!5! + x^14/7!7! - …

(the use of pre-formatted text is crucial here to see what's going on). Squaring the second:

 (cos x)²
= { Definition }
 (1 - x^2/2! + x^4/4! - x^6/6! + …)²
= { Distribution }
  1 - x^2/2!0! + x^4/4!0! - x^6/6!0! + …
    - x^2/0!2! + x^4/2!2! - x^6/4!2! + x^8/6!2! - …
               + x^4/0!4! - x^6/2!4! + x^8/4!4! - x^10/6!4! + …
                          - x^6/0!6! + x^8/2!6! - x^10/4!6! + x^12/6!6! - …

Note that the terms are set so that columns line up by exponent. Note also that the factorial dividends add up to the exponent. Now:

 (sin x)² + (cos x)²
=
      x^2/1!1! - x^4/3!1! + x^6/5!1! - x^8/7!1! + …
               - x^4/1!3! + x^6/3!3! - x^8/5!3! + x^10/3!7! - …
                          + x^6/1!5! - x^8/3!5! + x^10/5!5! - x^12/7!5! + …
                                     - x^8/1!7! + x^10/7!3! - x^12/7!5! + x^14/7!7! - …
+
  1 - x^2/2!0! + x^4/4!0! - x^6/6!0! + …
    - x^2/0!2! + x^4/2!2! - x^6/4!2! + x^8/6!2! - …
               + x^4/0!4! - x^6/2!4! + x^8/4!4! - x^10/6!4! + …
                          - x^6/0!6! + x^8/2!6! - x^10/4!6! + x^12/6!6! - …
= { Associativity of addition }
  1 - x^2/2!0! + x^4/4!0! - x^6/6!0! + …
    + x^2/1!1! - x^4/3!1! + x^6/5!1! - x^8/7!1! + …
    - x^2/0!2! + x^4/2!2! - x^6/4!2! + x^8/6!2! - …
               - x^4/1!3! + x^6/3!3! - x^8/5!3! + x^10/3!7! - …
               + x^4/0!4! - x^6/2!4! + x^8/4!4! - x^10/6!4! + …
                          + x^6/1!5! - x^8/3!5! + x^10/5!5! - x^12/7!5! + …
                          - x^6/0!6! + x^8/2!6! - x^10/4!6! + x^12/6!6! - …
                                     - x^8/1!7! + x^10/7!3! - x^12/7!5! + x^14/7!7! - …
= { Binomial theorem }
  1 - (x - x)^2/2! + (x - x)^4/4! - (x - x)^6/6! + (x - x)^8/8! - …
=
  1

Of course there are two fishy manouvers going on here. The first is the use of ellipses to denote the series. This is a notational convenience that allows me to show the full expansions for easy pattern matching of terms. These ellipses actually encode a rigorous inductive law defining each term, and could be replaced by summations. The second is the use of associativity to freely reorder the terms of an infinite series without regard to convergence, of which I can't say anything as I'm no mathematician but which I believe it can be made airtight. In any case, you can take this as a purely symbolic manipulation.