How is the induced electric field due to a changing magnetic field divergence free? How can we show that the divergence of the induced electric field is zero
 A: Mathematically, the divergence of a curl is always zero. Induced electric fields are represented by Faraday's Law of induction from Maxwell's Equations, which says the curl of the electric field goes like the (negative) time derivative of the magnetic field.  
The divergence of a curl is a little like finding 
$\vec A \cdot (\vec A \times \vec B)=0 $. 
Since $\vec A \times \vec B $ is perpendicular to $\vec A$, the dot product comes out to zero.
Physically, induced electric fields go in loops, and the divergence asks how many sources (beginnings or ends) of electric field lines there are. Since a loop doesn't have a beginning or end, the answer is zero.
Edit:
You might try looking at the electric field in terms of the potentials.  Here's an outline of something you might try.
$\vec E=-\nabla \phi-\frac{d \vec A}{dt}$
The induced electric field is just going to be
$\vec E=-\frac{d \vec A}{dt}$
$\nabla \cdot \vec A=C$
Where C depends on the gauge, but is time invariant.
So, 
$\nabla \cdot \frac{d \vec A}{dt}=\frac{d \nabla \cdot \vec A}{dt}=\frac{dC}{dt}=0$
