argument and atan/arctan As I have to works with sine voltages and currents, I often have to use complex numbers. 
I know that $\tan\arg(Z)=\left(\frac{\Re(Z)}{\Im(z)}\right)^{-1}$
but, how to prove that we need to add $ \pi$ in case $\Re(Z)\leq0$ ?
I know that $\arctan(x) + \arctan(\frac{1}{x})=\pm \frac\pi2, \forall x \in \mathbb{R}^*$ but I cant go any further.

That depends solely by the definition for the inverse function  $\arctan$ which is defined as the inverse of $\tan$ function on the interval $(-\pi/2,\pi/2)$. 
Therefore the derivation 
$$\theta=\tan(x/y)\implies x/y=\arctan \theta$$
is valid only for $\theta\in (-\pi/2,\pi/2)$ otherwise we need to correct by $\pi$.
