The number one thing that makes me not trust cops in any way if being questioned is that they are allowed by law to lie. I cooperate with law enforcement fairly often, but I would clam up if I ever thought for an instant that I was being asked questions where I could be a subject.
Meanwhile, if you lie, you go up for Felony Obstruction. Makes for quite an imbalance of power...sufficient to refuse to answer anything but name and DOB w/o counsel. You may think your answers are innocent, but you don't know what they know about the questions.
By this one particular we have been robbed of our presumption of innocence.
Logically, you can not presume someone is innocent while lying to them for the purpose of getting them to expose a crime.
It's like saying you offered someone a bribe because you presumed they wouldn't take it.