追う and 追い越す: following behind and overtaking
Compare running after someone with overtaking a walking friend, keeping an attempt to pass separate from a completed pass.
Following or chasing someone ahead
追う, read おう, means following or chasing a target. 先に走った人を追って走りました。 says that someone ran after the person who had run ahead.
逃げた犬を追いました。 says that someone chased a dog that had run away. Chasing does not itself establish that the dog was caught.
Moving ahead of the other person
追い越す, read おいこす, means overtaking. 歩いていた友達を追い越しました。 says that someone overtook a friend who was walking.
追い越した後で、足を止めました。 says that someone stopped moving after overtaking. た後で establishes that the pass preceded the stop.
An attempt and a completed pass
追い越そうとしましたが、前には出られませんでした。 says that someone tried to overtake but could not get ahead. Do not translate this attempt as a completed pass.
人を追っている describes following someone at present. It differs from 追い越しました, which reports a completed pass in the past.
Was the pass completed in 追い越そうとしましたが、前には出られませんでした。?
Show the answer and why
No. The person attempted to overtake but explicitly could not get ahead.