Wiring trailer brakes

I have a 97 sierra 1500 z71. I found a white wire at the back of the truck that works with the brake lights. It is a lighter guage wire, but could I use that for the brake wire on the the trailer brake controller intead of splicing into the brake pedal switch?

You could, conceivably, connect that white wire to a trailer brake controller instead of splicing at the brake switch, but that would involve about 15 extra feet of wire. Why not just splice into the wire at the brake switch and save yourself the hassle? Also, there may be a trailer brake connector under the dash if the truck was equipped with the factory tow package.

where abouts would that connector be under the dash?