Question

prove that electric potential is constant where electric field is zero

Answer 1

Electric field at any point due to the charge is defined as the negative gradient of potential at that point.
$E\hspace{0.17em}=-△V=-\frac{dV}{dr}$
gradient i.e. rate of change with respect to the position coordinates.

If E = 0
then $\frac{dV}{dr}=0$
i.e. rate of change in potential is zero.
i.e. V = constant
Hence proved.