complete a brawl without losing your hat as an LAPD Detective or Investigator.

You can achieve this achievement in any fight sequence, simply got LT to enter fight stance and use A button to punch and X to block, make good use of using X button when you see them about to punch which should make you dodge and then quickly press A and counter strike them which will deal more damage than normal.
For this achievement basically you can not be hit as if the bad guy punches you once your hat will fall off and you wont get it, the way i achieved this was just by simply pressing A as fast as i could and the bad guy couldnt react in time to even throw a punch back or even block, but bad guys later on will be better fighters so try and get this on one of the earlier fight sequences.