How do I make sure soldered and taped wires in a junction box at a light are properly insulated and protected

I noticed that the old tape on the 3 inches of wires soldered together in the juction box was brittle and falling off. Is the proper way to mend this is to just tape with electricial tape?

Pull off the old tape. If the connection still looks tight and good, re-tape it with modern vinyl electrical tape.

I have been an electrician for just about four years now, my answer to you is to have a licensed electrician come and re-splice your wires, unless your not afraid of electricity, if in that case, go to your local hardware store and buy some liquid electrical tape, turn off your power and take all the old tape off, try getting wire nuts on the wires and then coat the bundle in that liquid electrical tape......Usually you shouldn't have a bundle of wires taped together, so, if you wanted to be safe about it I would call a qualified electrician and have them come over and see if its up to code....

just retape it , there is a insulating tape , thats thicker, and stays soft, [like rubber]
may have to go to an elect supply to find it.

Turn the circuit breaker off that is wired to that fixture, take the old tape off, inspect the connection (look for kinked wires), if there are no kinked wires and the wires are soldered and looks strong reapply electrical tape.

If the soldered area looks bad cut the wires and get a wire nut and connect the wires and then use electrical tape.