[Sjw102]

Again not sure about an auto but on a manual there's a switch that, switches the reverse lights when reverse is engaged...

If this is the same on an auto, simple way to check is to unplug the plug to the switch and link the two wires together with a piece of wire or something and (with ign on ovs haha) see if the reverse lights are working, if so then it's the switch if not then the fault lies else where

 

[Sjw102]

Am not 100% sure but usually there screwed into the gearbox, so when reverse is selected in the box it pushes the switch in selecting it... But that's on a manual, auto boxes work different, but there could be a similar kind of thing