Direct Proof of De Moivre's Theorem

In §2.10, De Moivre's theorem was introduced as a consequence of Euler's identity:

To provide some further insight into the ``mechanics'' of Euler's identity, we'll provide here a direct proof of De Moivre's theorem for

*Proof: *
To establish the ``basis'' of our mathematical induction proof, we may
simply observe that De Moivre's theorem is trivially true for
. Now assume that De Moivre's theorem is true for some positive
integer
. Then we must show that this implies it is also true for
, *i.e.*,

Since it is true by hypothesis that

multiplying both sides by yields

From trigonometry, we have the following sum-of-angle identities:

These identities can be proved using only arguments from classical
geometry.^{3.8}Applying these to the right-hand side of Eq.(3.3), with
and
, gives Eq.(3.2), and
so the induction step is proved.

De Moivre's theorem establishes that integer powers of lie on a circle of radius 1 (since , for all ). It therefore can be used to determine all of the th roots of unity (see §3.12 above). However, no definition of emerges readily from De Moivre's theorem, nor does it establish a definition for imaginary exponents (which we defined using Taylor series expansion in §3.7 above).

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