2001 Cadillac Deville Fog lamps

Yes you can

You need to check the amp rating of the lamp that is already installed.
The switch rating should be higher or equal or you can use the relay you have in place.
Attach the Light circuit to the "Load" side of the relay. Run wire from power source's Hot supply to one of the relays load terminals (use fog fuse as power source, it is hot all the time). Connect the relays other load terminal to the light closest to it. Connect the remaining connection on the light to one of the connections on the other light. Connect the second light's remaining connection to the negative side of the power source.
Connect the switch. Run wire from a power source to one of the "Control" terminals on the relay. Connect the other terminal to the switch. Connect the remaining switch terminal to the negative of the power source (negative side of fog fuse).
Start car and test. If selected sources fuse blows, replace and try connecting to another source.

Hello scanssystems

Many kits come with pre-wired harnesses that make the electrical part of the job largely a plug-and-go affair. But you want to use the existing fog light.
Be sure to check state laws that regulate the number and power of auxiliary lights because, fog lights should be wired to work with the headlights in low-beam mode only. If you buy a kit for your vehicle, the instruction will right be there. It should not take so much time to install those and it should not cost so much, less than an hour. If you buy the whole kit, you can us the switch you have instead of using the switch that comes with the kit and everything should work fine.

How does the old fog light was control?
I mean how it was switched?

Install the new one and let me know if it work,
if it does not, let me know too.
So I can help you step by step.

Hello,
when you get it, take pic on the back side of it and the back side of the old one too and post it to me so we can continue.