Giving an exact description for $E_{R}:=\{\cos(z):\, R<|z|<\infty\}$ I have the following homework problem: 

For each $R>0$ prove $$A(0,0,\infty)=\{e^{z}:\, z\in A(0,R,\infty)\}$$
  and give an description of the set $$E_{R}:=\{\cos(z):\, z\in
 A(0,R,\infty)\}$$

Where $$A(z_{0},r_{0},r_{1}):=\{z\in\mathbb{C}:\, r_{0}<|z-z_{0}|<r_{1}\}$$
There is also a hint for the second part: if $\cos(z)=\omega$ for
some given $\omega\in\mathbb{C}$ then $e^{iz}$ must satisfy a certain
quadratic equation.
What I did:
I have proved the first part (which doesn't seem relevant for the
second part).
I have also used the hint: If 
$$
\cos(z)=\omega
$$
then 
$$
\frac{e^{iz}+e^{-iz}}{2}=\omega
$$
hence 
$$
\frac{e^{iz}\cdot e^{iz}+e^{iz}\cdot e^{-iz}}{2}=\omega
$$
Denote $t=e^{iz}$. Then 
$$
t^{2}-2\omega t+1=0
$$
Hence 
$$
t_{1,2}=\frac{2\omega\pm\sqrt{4\omega^{2}-4}}{2}
$$
$$
t_{1,2}=\omega\pm\sqrt{\omega^{2}-1}
$$
I also noticed that $e^{-iz}$satisfies the same equation hence one
of $e^{iz},e^{-iz}$ is $t_{1}$ and the other is $t_{2}$.
I don't see how this help me since 
$$
t_{1}+t_{2}=2\omega
$$
hence
$$
\frac{e^{iz}+e^{-iz}}{2}=\frac{t_{1}+t_{2}}{2}=\omega
$$
so this doesn't give me any new information. 
This is where I am stuck, can someone please help me continue this
exercise ? 
 A: Solution 1
Let $f(z)=e^{iz},$ so $f(A(0,R,\infty))=A(0,0,\infty)$
Also, $g(z)=\frac12(z+\frac1z)$
Then $\cos z=\frac12(e^{iz}+e^{-iz})=(g\circ f)(z)$
Where foes $g$ map $A(0,0,\infty)?$
Well, generally $g$ maps the points $\{|z|>1\}$ conformally to $\mathbb C\setminus [-1,1]$
(See Bak-Newmann , Riemann Map chapter)
.
Also, the points of the circumference $|z|=1$ are mapped to the interval $[-1,1]$
Therefore, the image of $|z|\geq 1$ is the whole plane, which implies that the image of $A(0,0,\infty)$ is the whole plane.
Solution 2 (using the hint)
You have proved that the image of $A(0,R,\infty) $ through $e^z$ is $A(0,0,\infty)$. The same holds if we replace $e^z$ with $e^{iz}$.
Now we will prove that the image of $A(0,R,\infty)$ through $\cos z$ is the whole plane. 
Take a $w\in \mathbb C$. We need to prove that there exists $z\in A(0,R,\infty)$ such that $$\cos z=w\iff$$
$$e^{2iz}-2we^{iz}+1=0\iff $$$$t=e^{iz}, t^2-2wt+1=0$$
Therefore $t=\dfrac{2w\pm\sqrt{4w^2-4}}{2}=w\pm\sqrt{w^2-1}$
This proves already that for a given $w\in\mathbb C$, you can find a $t\in\mathbb C$ which is not zero, therefore it belongs in $A(0,0,\infty)$ . Now, obviously(from the first part of the exercise), for a given $t\in A(0,0,\infty)$ we can find a $z\in A(0,R,\infty)$ such that $e^{iz}=t$ and the proof has finished.
