This monstrosity is in my attic. There is one attic light. 
There is a legitimate wire from the light junction 'fixture' to a junction box elsewhere. (It's the white wire that you see behind the light.)
HOWEVER, there is another wire coming out of the light fixture that goes to a transformer…which then splits into two loose wires that have a suspect connection using electrical tape. At places along the two wires there seems to be some wear and tear with the insulator. I probably should use some electrical tape.
I didn't trace the wires to the end but they seem to go towards the front door where there is a doorbell and a porch light. (The roof pitches down near there and there's hardly any space to see where the wires go)
I used a voltage tester and it beeps to indicate that it is hot.
What should I do with this wire? I want to cut it. How do I figure out what it's connected to?
Do I turn off the breaker to the attic light and then disconnect that transformer? If I do that then the device that is connected to it should stop working.
Best Answer
EEK! Kill it with fire, before it kills you with fire!
I'm referring to the breakable Edison screw base bulb that is on the floor and underfoot, ready to be tripped over, tearing the porcelain base socket off and exposing you to 120V. UGH. Remember, if you bust that bulb, you will have live 120V inside broken glass right next to grounds, and there you are groping around in the pitch black! Unacceptable!
Here's what I would do.
Get a modern transformer that's built into a junction-box lid, like this one. Not for this junction box - this box gets a blank lid, so you can climb across it without touching anything electrical.
In the (illegal) hole the transformer wiring went through, I'd get some EMT conduit and extend it along that joist, toward the eave, as far as practical before the clearances get too close to work. I'd install another junction box there, screw it to the joist with 2 screws. I'd fasten the EMT to the joist rather firmly with lots of clamps so you can climb/step across it or put stuff on it. There'll be a 1/4" or so gap between EMT and joist, the usual practice is 4 subtle bends, I'd just put shims under it.
In that new box I'd install the new "in the lid" transformer.
Out of the side of that junction box, I'd go EMT, flexible conduit or armored cable to a suitable location for LED lights. I'd put two there, selecting models unlikely to be damaged from rough handling of boxes etc. in the attic. Two, so I can use cheapies and still have light to fix one when it fails.
Ground wires needed also, I'm not showing the grounds since wherever you use metal EMT, that can be the ground. I also overlooked a light switch, I don't know how the attic light turns on now. If needed you can run some flexible armored cable over to a convenient location for a switch.
You could also relocate the existing box to be on the sides of the rafters instead of the top, but that's more than I wanted to get into here.
By the way, the transformer is the "ridiculously easy" part of this. The REAL work, and real win, is fixing that dangerous lighting situation and making your attic space a lot more usable. Besides, wireless doorbells look cheap IMO and hurt your resale. If you've got a real one, it's not hard to keep it.