  We employ magnetocapacitance measurements to study the spectrum of a double
layer system with gate-voltage-tuned electron density distributions in tilted
magnetic fields. For the dissipative state in normal magnetic fields at filling
factor $\nu=3$ and 4, a parallel magnetic field component is found to give rise
to opening a gap at the Fermi level. We account for the effect in terms of
parallel-field-caused orthogonality breaking of the Landau wave functions with
different quantum numbers for two subbands.
