I have never owned a Purple Tang, however my Sailfin Tang and Gold Stripped Maroom clown are the two most agressive fish in my tank. You have the Tang in place so it is making itself at home and will believe the entire tank to be "Its space" you may find that you will need to rearrange the rock to add the clown and keep them from aggitating each other. That was how I finally got the Sailfin and Clown to quit fighting.
That is if the Purple Tang is as aggressive as my Sailfin. The Sailfin also will eat any invert it can, seems to never be full, when I move some algae from the sump/refuge up to the main tank it finishes it off within 24 hours. The shrimp hide and are quick enough but they do not come out into full view during the day for this reason.