The 3/8 " end of the 1/2" to 3/8 " faucet connector neither fits a 3/8" pipe nor a 1/4" pipe. Right now I have to find a connector which connects the 3/8" end of the faucet connector to 1/4" pipe. Which connector should I be looking for ?
Whoa... remove everything you just installed, including those valves.
- the valves you used have garden hose thread outlets, and you have incorrectly screwed pipe thread fittings to them. It will leak, if not now then in the future. Those valves are commonly used to supply clothes washing machines which use supply lines with garden hose thread.
Replace the valves with dual-outlet angle stop valves, which come with the proper 1/2" pipe thread inlet and outlets that will fit your 3/8" compression thread flex supply and your 1/2" pipe thread flex supply:
Don't use those black iron fittings; they will corrode. You won't need them anyway if you purchase the correct stop-valves.