I'm not sure to be honest. The approach I have taken is as follows:
= |1/2 (e^(1+iz^2)+e^-(1+iz^2)) / 1/2(e^iz+e^-iz)|
= e^1+|z^2|+e^1+|z^2| / |e^z|+|e^z|
= e^1+|z^2| / e^|z|
When |z|< 3, this equates to e^7
It looks right but I can't find a way to check this.