Making deliberate false claims in an effort to coerce someone to do something they don’t want to do is the definition of EXTORTION and it is a FELONY everywhere. NO SPECIAL RIGHTS, Government Agents that do this belong in PRISON.
It is a slippery slope, though. A guilty criminal would not ordinarily confess to a crime, so telling a guilty criminal that they arrested someone else for the crime in order to get the guilty person to say something may not be entrapment because the underlying crime was committed without furtherance by the police.
On the other hand, telling an innocent person that something is not a crime when it is, and then arresting that person for doing it, is entrapment.
-PJ