Answer
No
Work Step by Step
Let us consider a vector field $G$ such that $div [curl (G)]=0$
consider $F=A i+B j+C k$
$div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$
We are given that $curl G=\lt x \sin y, \cos y, z-xy \gt$
$div[curl(G)]=\sin y-\sin y+1 $
This implies that $div [curl (G)]=1 \ne 0$
Thus, there does not exist such a vector field $G$.
