Suppose the following wave has amplitude 1.
$y(x,z,t)=Ab\cos(px)e^{iwt+ikz}$
Find a relationship between $A$ and $b$. Then rewrite $y$ in terms of that relationship.
My thoughts.
The amplitude is the largest value $y$ can take. So $|Abcos(px)e^{iwt+ikz}|=|Ab|=1$.
Is that all that is required?
Honestly, yes. In fact, $|Ab|=1$ means that $Ab= 1$ or $Ab=-1$, so either $$y(x,z,t) = \cos(px) {\rm e}^{{\rm i}\omega t+ {\rm i}kz} \quad \mbox{or}\quad y(x,z,t) = \cos(px) {\rm e}^{{\rm i}\omega t+ {\rm i}kz}.$$I wonder why you were even asked this --- the expression for $y(x,z,t)$ being more complicated than the usual $y = A\cos(Bx-C)+D$ people see in precalculus classes would suggest something more is going on, but oh well.