  In this paper we derive a gauge fixing identity by varying the covariant
gauge fixing term in $Z[A,J,\eta, {\bar \eta}]$ in the background field method
of QCD in pure gauge. Using this gauge fixing identity we establish a relation
between $Z[J,\eta,{\bar \eta}]$ in QCD and $Z[A,J,\eta, {\bar \eta}]$ in
background field method of QCD in pure gauge. We show the validity of this
gauge fixing identity in general non-covariant and general Coulomb gauge
fixings respectively. This gauge fixing identity is used to prove factorization
theorem in QCD at high energy colliders and in non-equilibrium QCD at high
energy heavy-ion colliders.
