De Moivre's formula
Proof by induction

The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer n, call the following statement S(n):

For n > 0, we proceed by mathematical induction. S(1) is clearly true. For our hypothesis, we assume S(k) is true for some natural k. That is, we assume

Now, considering S(k + 1):

See angle sum and difference identities.

We deduce that S(k) implies S(k + 1). By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, S(0) is clearly true since cos(0 x) + i sin(0 x) = 1 + 0 i = 1. Finally, for the negative integer cases, we consider an exponent of − n for natural n.

The equation (*) is a result of the identity

for z = cos nx + i sin nx. Hence, S(n) holds for all integers n.
combination:repeat_prompt