Corrected entry: Kate uses her real first name while she is working undercover to catch whomever is killing people from the nightclub. According to a friend in law enforcement who checked her facts with officers from two different agencies, this would never be allowed.
Corrected entry: Angel decides to talk to Kate about the demon and calls the Los Angeles Police Department for her. At this point in the series, he hasn't been told her last name, and Kate could be short for Katherine, Karen, or Kathleen; yet despite the fact that the LAPD employs thousands of officers, several of whom must have one of those first names, he is put through to the right Kate. He can't have read her name off her badge from across the room in the 4 seconds she had it out either, because vampires in the Buffy Universe can see well In The Dark, however, they don't have telescopic vision. (00:33:40)
Correction: Angel could have described her to the person on the other end of the phone & got through that way.
Correction: Well, Kate is a pretty common first name, so as long as she wasn't using her real last name, nobody would know that she is a cop.