If either the client or provider asks the other "are you a member of any branch of law enforcement?"  Can a LE member answer no?  Or do they have to answer yes?

purpose towards capturing a perp.

Also, anything you say can and will be used against you, so don't tell them anything.

(still not a lawyer)

They do not have to say yes.

Entrapment is legal in the united states

Entrapment is NOT legal. But this is NOT entrapment. Entrapment is hard to prove. It is LE causing a person to commit a crime they would not have but for the actions of LE. On this logic, one could get away with murder if you asked a witness if they were a cop first, and they lie and say no. Still illegal to kill someone

3 things to prove that entrapment  has occurred, but you have to prove that all 3 occurred in order to win your case
  1.  The idea of committing the crime came from law enforcement officers, rather than the defendant.
  2. The law enforcement officers induced the person to commit the crime. Courts have traditionally maintained a high burden of proof for inducement. Simply affording the defendant the opportunity to commit the crime does not constitute inducement. For inducement to be proved, officers must have used coercive or persuasive tactics.
  3. The defendant was not ready and willing to commit this type of crime before being induced to do so. If an undercover cop bought cocaine from a person carrying a kilogram of the drug, the seller could not plead entrapment, even if coercion were involved in the sale, since his intent to sell was clear. Most courts also allow a defendant's predisposition to be demonstrated through prior conduct or reputation.