It's sometimes called "trickery". Although I am not familiar with the law in your state, my familiarity with both my state's law and that of the Federal Courts tells me that the police tactic you've described would be considered lawful. I will wait with you for agreement or correction with others who are knowledgeable with your state's conclusion on this issue. In any event, I see no impact on your son's case.
Mr Jones is correct. Unless there is a specific Louisiana law that prohibits the police conduct which you describe in your question, what the police did was legal. You may feel betrayed but their job is to solve crimes. I would suggest you speak to a criminal defense lawyer in the area (if for no other reason than peace of mind) to find out the answer to your question. Good luck.
I would agree with my two colleagues. I would only add that this is a perfect example of why you should never make a statement to law enforcement without first consulting an attorney. It is rare that an uncounseled statement would ever help a persons situation. Having said that, I do not believe it will be of any consequence in your situation.