Existence and understanding holomorphic root functions

Question

I have to prove that
a) for all $|z|<\frac{\pi}{2}$ exists a root function of the function $f(z)=\cos(z)$, which means there is a holomorphic function $g:\mathbb{D}\to \mathbb{C}$, such as $g(z)^2=f(z)$ for all $z$, $|z|<\frac{\pi}{2}$ ($\mathbb{D}$ is the unit disc) and
b) There exists no holomorphic function $\omega:g\to \mathbb{C}$, such as $\omega(z)^2=z$ for all $z\in G:= \{z\in \mathbb{C}:1<|z|<2\}$.
With a) I think it follows straight from my lecture notes: Since the circle with radius $\pi/2$ is a simply connected domain and $\cos$ is holomorphic on it and has no zeros there follows the existence of a holomorphic root function. But this seems a bit to easy for me.
With b) I have no clue to show the nonexistence. Maybe the idea is to show that there is no holomorphic logarithm on the Annulus?