I'd generally agree that the police shouldn't be able to lie to a detained suspect, or at least that such deception should be more constrained than it is in the context but, in general, the law as practiced hinges on the question, "does police deception render a detained suspect's confession or statements involuntary?"

If yes, then those confessions or statements could become inadmissable (i.e. the suspect was effectively compelled to testify against himself, in violation of the 5th Amendment). When police combine deception with coercion or threats, the answer to that question leans toward yes, but it will depend on the juristiction and the judge and is often assessed on a case-by-case basis. Police in any given juristiction probably have a reasonably good sense (from their own accumulated experience and local training) of what kind of deception results in the produced evidence becoming inadmissable in the courts where the defendant will appear.