Math & Physics Problems Wikia
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Problem

First prove the formula . Then discover the beautiful identity .

Hint: Use the Maclaurin series of exponential and sinusoidal functions.

Complex unit circle.png

Solution

Begin with the Maclaurin series of .

Replace with , where is the imaginary unit.

The powers of follow a pattern:

, for
, for
, for
, for

Thus,

This shows that .

Setting yields

.

Therefore,

.
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