Police are absolutely allowed to lie during an interrogation. I’ve seen it done multiple times. The most disturbing interrogation I’ve ever seen was actually the one where a teenager murdered his own father while under the influence of magic mushrooms. He was taken into the interrogation room shortly after the murder but enough time had elapsed that he was sober and just starting to understand the gravity of what he had done. The police sent in a soft-voiced female who spoke in a very caring and patient way. 99/100 people who watch that interrogation video would say that the woman was a therapist or councilor of some kind. Everything she asked and stated appeared to concern the wellbeing of this kid and the resolution of the terrible even that had taken place. In reality, everything she did was a cunning ploy with some kind of hidden goal. The entire interaction was calculated to extract information from him, very specific information, to be used against him in court. Every tangent of questioning she went on, seemingly random to the untrained eye, was highly specific in its value in a legal context.
An interrogation like this is not an example of outright lying, which is legal and commonly practiced, but it is much more sinister. It is an example of an interrogator assuming another identity and drawing the suspect into a false sense of warmth and security in order to make the suspect incriminate himself. To me, it is the same as having a conversation with your mother after a traumatic event, only to see her rip her own face off, a mask, to reveal the grotesque face of a police interrogator, and then going to jail for what you disclosed. Absolutely terrifying and dystopian. You absolutely must be proactive and protect yourself in this world. Even in “free” countries like the USA. https://youtu.be/rBpDHJIwcUk?t=1495
TLDR: the police are allowed to lie to you https://youtu.be/_WnhP91NJeU?t=1600