Question

using one of the maxwell equations show that a time-dependent magnetic field cannot exist without an electric field?

Answer 1

You can use Faraday's Law:\[\nabla \times \vec{E} = \frac{\delta \vec{B}}{\delta t}\]
If you make the electric field a null vector, you can just integrate by time (the left side is a constant vector) and you realize that you will still have a null vector times t. No magnetic field exists.