Oh Angels are very territorial. I tried to put a Six Line Wrasse in my tank that had (at the time) a Goby, Maroon Clown, and Pygmy Angel. As soon as the Six Line hit the water after acclimation my Angel came flying out of the rocks and nearly blew the Wrasse right out of the water. He kept chasing it up to the top of the tank in the back, as if to make him stay there. The Angel wanted nothing to do with that Wrasse being in the LR. The Wrasse wouldnt eat and got incredibly stressed and died the next day.
Since then I have added a Dwarf Lion and then a Valentini Puffer 5 weeks later. The Angel didnt mess with either of those guys LOL.
I agree with the above, the best way to "teach the Angel a lesson" is to remove it, along with the Tang, and re-arrange some rock etc. Put that new fish in and let it acclimate for several days, then return the troublemakers to the tank lol.