How to go from $\frac{1}{1+2j}$ to $\frac{1}{5} - \frac{2}{5}j$, where $j^2=-1$?

Question

I am reading a book (DSP First), or mainly skipping through the pages trying to solve various exercises. At some point I came across this

How exactly did we go from the second to the last step?

Answer 1

$$(1+j2)(1-j2)=1^2-j^22^2$$

and $$j^2=-1$$

Answer 2

Notice, in general when $z\in\mathbb{C}$:

$$\frac{1}{z}=\frac{\overline{z}}{z\overline{z}}=\frac{\overline{z}}{|z|^2}=\frac{\Re[z]-\Im[z]i}{\Re^2[z]+\Im^2[z]}=\frac{\Re[z]}{\Re^2[z]+\Im^2[z]}-\frac{\Im[z]i}{\Re^2[z]+\Im^2[z]}$$

Where $\overline{z}$ is the complex conjugate of $z$ and $\Re[z]$ is the real part of $z$ and $\Im[z]$ is the imaginary part of $z$.

So:

$$\Re\left[\frac{1}{z}\right]=\frac{\Re[z]}{\Re^2[z]+\Im^2[z]}$$ $$\Im\left[\frac{1}{z}\right]=-\frac{\Im[z]}{\Re^2[z]+\Im^2[z]}$$