OK, from the pic, it looks like you have the rather common shutoff valves that are 1/2 FPT on one end and 3/8 compression on the other with plastic lines going up to the faucet. There are a couple of different ways you can do this.
One) You can get a fitting that will install behind the shutoff valve and your
RO supply line will connect to this fitting. I don't like to do it this way because I like to keep the
RO supply on the other side of the existing shutoff valve...that way you can turn the water off using your existing shutoff valve.
Two) You can get a compression tee in the appropriate size (3/8x3/8x1/4 I think). Cut the plastic line going up to the cold water side of the faucet, connect it back together with the compression tee, and tie your
RO supply line into the third opening in the tee (the 1/4" connector). This will allow you to continue to use the existing shut off valve to cut off the water in the event of a leak.
Three) Remove the shut off valve from the wall plumbing and screw on a 1/2 tee with one open end pointing straight out and one pointing up. In the end that's pointing out, screw in a 1/2 close nipple (using teflon thread sealer tape of course) and reinstall the shut off valve there. Purchase a shutoff valve that is 1/2 MPT (male pipe thread) on one end and 1/4 compression on the other (You'll probably find this in the brass fittings section at a hardware store). Screw this in the top of the tee and connect your
RO supply line here. This will let you shut off the
RO supply at the source and you won't have to cut into the faucet plumbing. A piercing valve is not an option if you have the plastic supply lines going up to the faucet.
HTH.
