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. OUTLINE Base case 1+1 =/= 1 follows directly from (2). Inductive step ...

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