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.
ixleads 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: