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Zeroes of trigonometric-like function

Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ALL zeroes of the corresponding $f(z)$ belong to coordinate axes in $z$. … . $$ Obviously the line $b=0$ is not in $D$, as zeroes of $\cos(z)\cosh(az)=0$ belong to axes $Re z=0, Im z=0$. Problem: define geometrically the set $D$. …