Hmm, I'm surprised than no one mentioned de Moivre's formula: $$(\cos \phi +i\cdot \sin \phi)^n= \cos (n\phi) + i\cdot\sin (n\phi)$$
It is an excellent theorem to prove where we must also use the addition theorem for sin and cos.
Prove that, for all x in N, we have x+1 =/= x. (Requires use of proof by contradiction.)
Use only the following properties of addition on N:
Addition is closed on N.
x+1 =/= 1 for all x in N.
If x+1 = y+1, then x=y for all x, y in N.
The Principle of Mathematical Induction.
1+1 =/= 1 follows directly from (2).