Q:

Show that $z^{-1} = \frac{\bar z}{|z|^2}$

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!

A:

HINT: $$\frac1z=\frac{1}{x+iy}=\frac{1}{x+iy}\cdot\frac{x-iy}{x-iy}=\dots ?$$

A:

$\frac{1}{z} = \frac{\overline{z}}{\overline{z}} \frac{1}{z} = \frac{\overline{z}}{|z|^2}$ (since $z \overline{z} = |z|^2$).

