Making the denominator of a certain fraction real

[TensorCalculus]

Homework Statement

Show that, for |r|<1, $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$

Relevant Equations

$$e^{ix} =\cos(x) + i\sin(x)$$

Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers:
Show that, for $|r|<1,$
$$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$

My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above:
$$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$

The sum of this series is just:
$$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$

I'm having some trouble trying to figure out what to multiply the numerator and denominator by in order to make the denominator real (and, thus, be able to split it into real and imaginary parts). If it were just $\frac{(e^{ix})^n-1}{e^{ix} - 1}$ I could just multiply by $e^{-ix/2}$ which would make the bottom just 2x a sine function... but you can't do that for this one because of the r and I'm not sure what to do: I tried expanding $e^{ix}$ out etc but I couldn't find any way to make the denominator real then either. I'm not really sure what to do...

Any hints as to how I can split this into the real and imaginary parts would be much appreciated thanks!