I'm stuck on this question, I have a feeling the answer is very straightforward but I just can't figure it out.
Question: Considering $z= x + iy$, show that: $$z^{-1} = \frac{\bar z}{|z|^2}$$
So far this is what I have: $\bar z=x-iy$ and $|z|^2= x^2 + y^2$
Therefore: $$\frac1{x+iy}=\frac{x-iy}{x^2 + y^2}$$
Where do I go from here? Thanks!