This can be verified in a particularly simple way.

The series representation of the exponential function:

is especially neat because each term in the series is the derivative of the term following, and the result of that is:

Which is, indeed, one definition of this function.

Substitution of

`ix`

leads to a simple shift in the pattern:repeating with a period of 4. Powers of x like:

But remembering the series for sine and cosine, and multiplying the former by i:

Adding:

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