Cops should NEVER be allowed to lie outside of specific, warrant backed undercover operations. I will never understand when it became ok for those charged with enforcing the law to lie without shame.
Lying, by itself, is not a crime. So why should they not be able to lie?
Generally speaking, they may not commit a crime in their pursuit to solve or prevent crimes. Police don't have diplomatic immunity. Again generally speaking, the same laws that apply to you also apply to them too.
What they are NOT allowed to do is commit crimes. They might be able to lie, but not commit fraud or "entrap", which basically means to talk or fool someone into doing something illegal they would not normally do.
In one state I know of, the police were trying to have it both ways. They claimed they were "always on duty", even though of course they were only being paid for 8-hour shifts. They wanted to be able to do whatever they wanted when not on shift, yet still carry their guns and make arrests, and act like police, any time of the day or night, whenever they wanted.
The State said no, if they were "always on duty" they'd have to be job-insured 24 hours a day, and get "on-call" pay in addition to their regular wages. Since the State wasn't about to do that, it declared officially that police were only "really" police when they were on their paid shift. Any other time, they can only make citizen's arrests, just like everybody else. Also, as a result, the few exemptions the police got for firearms carry are only in effect when they're on paid duty. All firearms laws in the state apply to off-duty police in exactly the same way they do to everyone else. The police really howled about that one.
Prior to the State "cleaning house", the police were also busting prostitutes by paying them and having sex with them, then arresting them and taking the money back. The state said "no way". No more hands-on for the police. They howled about that one too.