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 *e* 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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