An example for a calculation where imaginary numbers are used but don't occur in the question or the solution. In a presentation I will have to give an account of Hilbert's concept of real and ideal mathematics. Hilbert wrote in his treatise "Über das Unendliche" (page 14, second paragraph. Here is an English version \- look for the paragraph starting with "Let us remember that we are mathematicians") that this concept can be compared with (some of) the use(s) of imaginary numbers. He thought probably of a calculation where the setting and the final solution has nothing to do with imaginary numbers but that there is an easy proof using imaginary numbers. I remember once seeing such an example but cannot find one, so: Does anyone know about a good an easily explicable example of this phenomenon? ("Easily" means that enigneers and biologists can also understand it well.)

Once nice example is the sum

$$ \cos x + \cos 2x + \cos 3x + \cdots + \cos nx $$

This can be worked out using trigonometric identities, but it turns out to be surprisingly simple with this neat trick:

$$ \sum_{k=1}^n \cos(kx) = \sum_{k=1}^n\mathscr{Re}\\{e^{ikx} \\} = \mathscr{Re} \sum_{k=1}^n e^{ikx} = \mathscr{Re}\left\\{ \frac{e^{i(n+1)x} - e^{ix}}{e^{ix} - 1} \right\\} $$

because the sum turns into a geometric series. (computing the real part to get an answer in terms of trigonometric functions is not difficult, but is a little tedious)