Answer
$ \frac{N_v}{e^{-(E_V - E_F)/kT} + 1} = \frac{N_c}{e^{E_c - E_F/kT} + 1} $
Work Step by Step
$N_{ev} = N_vP(E_v) = \frac{1}{e^{E_v - E_F/kT} + 1}$
Since there are total of $N_V$ states inn valence band, the number of in the valence band is
$N_{hv} = N_V - N_{ev}$
$N_{hv} = N_V [ 1 - \frac{1}{e^{E_V - E_F/kT} + 1}]$
$N_{hv} = \frac{N_v}{e^{-(E_V - E_F)/kT} + 1}$
The number of electron in the conduction band is
$N_{ec} = N_cP(E_c) = \frac{N_c}{e^{E_c - E_F/kT} + 1}$
take $ N_{ec} = N_{hv} $
we get
$ \frac{N_v}{e^{-(E_V - E_F)/kT} + 1} = \frac{N_c}{e^{E_c - E_F/kT} + 1} $
in which $-(E_V - E_F) = \Delta E_v $ and $(E_c - E_F) = \Delta E_c$
