Q:

Applying representation $f(z)=w_0+\zeta(z)^n$ to $\cos z$ (from Ahlfors' Complex Analysis).

I'm working on an exercise which asks to apply the representation $f(z)=w_0+\zeta(z)^n$ to $\cos z$ with $z_0=0$, and determine $\zeta(z)$ explicitly. (This is from Ahlfors, page 133, ex. 3 by the way).
Now $\cos(0)=1$, so $w_0=1$. Also, $\cos z-1$ has a zero at $z_0=0$ of order $2$. So I can write $\cos z-1=z^2g(z)$ with $g(z)$ analytic at $z_0$, and $g(z_0)\neq 0$. Taking $z^2g(z)=\zeta(z)^2$, I just find $\zeta(z)=\sqrt{\cos z-1}$, so
$$
\cos z=1+\sqrt{\cos z-1}^2.
$$
I'm worried, have I done this correctly? I don't feel like I've done anything of substance, so I think the exercise might have been asking for something else. 
I think Ahlfors is referring to a theorem stating

Suppose that $f(z)$ is analytic at $z_0$, $f(z_0)=w_0$ and that $f(z)-w_0$ has a zero of order $n$ at $z_0$. If $\epsilon>0$ is sufficiently small, there exists a corresponding $\delta>0$ such that for all $a$ with $|a-w_0|<\delta$ the equation $f(z)=a$ has exactly $n$ roots in the disk $|z-z_0|<\epsilon$.

Later discussion states that under the assumption of this theorem, we can write
$$
f(z)-w_0=(z-z_0)^ng(z)
$$
where $g(z)$ is analytic at $z_0$ and $g(z_0)\neq 0$. Choose $\epsilon>0$ so that $|g(z)-g(z_0)|<|g(z_0)|$ for $|z-z_0|<\epsilon$. In this nbhd is's possible to define a single-valued analytic branch of $\sqrt[n]{g(z)}$ which we denote by $h(z)$. We have thus
$$
f(z)-w_0=\zeta(z)^n,\qquad \zeta(z)=(z-z_0)h(z).
$$
Since $\zeta'(z_0)=h(z_0)\neq 0$, the mapping $\zeta\zeta(z)$ is topological in a neighborhood of $z_0$. 
I think the implied representation is related to the conventions here.

A:

HINT: Note that $1-\cos(2z)= 2\, \sin^2(z)$.

