S = {e

s = ΣS = Σ

S' = {e

Let s' = ΣS'. s = s'. But s' = e

(1 − e

e^{ix} = cos(x) + i sin(x).

0 = Σ_{0 ≤ k < n} e^{2πik/n}

= Σ_{0 ≤ k < n}(cos(2πk/n) + i sin(2πk/n))

= (Σ_{0 ≤ k < n}cos(2πk/n)) + i(Σ_{0 ≤ k < n}sin(2πk/n)).

Thus:
Σ_{0 ≤ k < n}cos(2πk/n) = Σ_{0 ≤ k < n}sin(2πk/n) = 0.

Q.E.D.

This is unsuitable for a trigonometry class that has not been introduced to complex numbers. If I were teaching trigonometry I would give an unrigorous geometric proof.

When I learned that e^{ix} = cos(x) + i sin(x), and why, I discarded by previous vague definitions of sin and cos in favor of their definition by taylor series and never looked back.