Solution
= Solution
The first identity in part (ii), with $F$ and $G$ interchanged, says
$$
\nabla\cdot(G\times F)
+=F\cdot(\nabla\times G)-G\cdot(\nabla\times F),
$$
so the proposed identity is missing its final term. For a concrete counterexample, take
$$
G=e_x,
\qquad F=ye_z.
$$
Then $\nabla\times G=0$, so the proposed left side is zero, whereas
$$
G\times F=-ye_y,
\qquad
\nabla\cdot(G\times F)=-1.
$$
\b[Thus the identity is false.]