Step 1: Use continuity of current:
Current must be continuous in a series circuit. In the wire it is the conduction current. In the gap it must be carried by something else, and that is the displacement current.
Step 2: Write the relation:
For a parallel plate capacitor, $I_d = \varepsilon_0 \frac{d\Phi_E}{dt} = \frac{dq}{dt}$. The right side is the rate at which charge arrives on the plates, which is $I_c$.
Step 3: Check the source:
This result uses only charge conservation, not the shape of the source voltage. So it is valid for a sinusoidal source and for a battery.
Step 4: Conclusion:
Hence the two currents are the same for either a.c. or d.c.
Final Answer:
Equality of the two currents holds for both kinds of source.
\[ \boxed{\text{either a.c. or d.c.}} \]