Or basically anything that isn't a positive integer.

So I can prove quite easily by induction that for any integer n>0, De Moivre's Theorem (below) holds.

If ##\DeclareMathOperator\cis{cis} z = r\cis\theta, z^n= r^n\cis(n\theta)##

My proof below:

Spoiler

Let P(n) be the statement ##z^n = r^n\cis(n\theta)##

Prove true for n = 1:
##z^1 = z = r\cis\theta = r^1\cis(1\theta)##

Assume true for n = k:
##z^k = r^k\cis(k\theta)##

Prove true for n = k + 1:
##z^{k+1} = z^k \times z\\
= r^k\cis(k\theta) * r\cis\theta\\
= r^{k+1}\cis(k\theta)\cis\theta\\
= r^{k+1}(\cos(k\theta) + i\sin(k\theta))((\cos\theta + i\sin\theta)\\
= r^{k+1}(\cos(k\theta)\cos\theta + \cos(k\theta)i\sin\theta + i\sin(k\theta)\cos\theta - \sin(k\theta)\sin\theta)##
then by compound angle identities
##= r^{k+1}(\cos(k\theta + \theta) + i\sin(k\theta + \theta))\\
= r^{k+1}\cis((k+1)\theta)##

However I struggle to do this with negatives. I tried the simple case of

##\frac{1}{r\cis\theta}## and from there, try to get to ##\frac{1}{r}\cis(-\theta)##, but I've only managed to do that by replacing the numerator of 1 with ##r\cis0##, which is equivalent, and then from there using De Moivre's Theorem to get an argument of ##-\theta##, which is... not very useful considering De Moivre's Theorem is what I'm trying to prove in the first place.

I just can't manipulate it into a form that would allow me to use compound angle identities like I could with positive numbers. Any ideas?

##z^{-1}##, but my proof in the OP only works for positive integers. I'm trying to use -1 as a base case to write a proof for negative numbers, but for some reason I cannot work out how to apply a similar strategy (expand then use compound identities)

It doesn't matter if θ is positive or negative, what matters is its multiplier, which is n.

Well... they are equal, but thus far my understanding of it is only intuitive, until I can prove it, which is what I'm asking for help with

Ok, so if ##z=a+ib##, where ##a, b## are real, how would you compute ##\frac{1}{z}=\frac{1}{a+ib}##?
Hint: multiply numerator and denominator by ##a-ib##.

Once you have proven that ##\frac{1}{\cis \theta}=\cis (-\theta)##, you can easily transform the formula with a negative ##n## into a formula with the positive ##-n##. And then you can use what you already have proven.