Problems solving argument?

$Arg\left (\frac{i\omega}{2-i\omega} \right )=Arg(i\omega)-Arg(2-i\omega)$

Solution:
$\pi-arctan\left (\frac{\omega}{2} \right )$
$\omega>0$

My attempt:

$Arg(i\omega)=\frac{\pi}{2}$

$Arg(2-i\omega)=-arctan\left (\frac{\omega}{2} \right )$

$Arg(i\omega)-Arg(2-i\omega)=\frac{\pi}{2}-\left (-arctan\left (\frac{\omega}{2} \right ) \right )=\frac{\pi}{2}+arctan\left (\frac{\omega}{2} \right )$

Can somebody explain why the solution differs from my solution?
I have tested Arg(iw) with any number w>0 and they all give me pi/2?

Also I don't understand how they got the subtraction when mine ends up being a sum?