Answer
$curl (curl F)=grad(div F) -\nabla^2 F$
Work Step by Step
We need to prove that $curl (curl F)=grad(div F) -\nabla^2 F$
This implies that
$curl (curl F)=\nabla \times (\nabla \times F)$
or, $curl (curl F)=\nabla (\nabla \cdot F) -F (\nabla \cdot \nabla)$
or, $curl (curl F)=\nabla (div F) -F (\nabla^2)$
Hence, $curl (curl F)=grad(div F) -\nabla^2 F$
