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This follows from e^(ix) = cos(x) + i sin(x)! I'm currently reading the Qiskit quantum computing textbook, and the appendix on linear algebra has a demonstration: https://qiskit.org/textbook/ch-appendix/linear_algebra.html

And that follows from De Moivre's formula, (cos(x)+ i sin(x))^n = cos(nx) + i sin(nx).

Also an amazing way are Taylor polynomials, this article explains the process of thought really well:

Yeah, in the link I posted the formula is derived via Taylor (or Maclaurin) series, but the explanation in your link is great. Thanks for sharing!