I'll try and be a little more opinionated... For N1 I would definitely advocate for medics and exorcists to not use their power. I think the detective should use his power, as the benefit is great and the odds of hitting cult are small. To be safe, I'd prefer the cop to target detective or detective's target, to decrease chance of getting culted.
Math for brain pleasure:
Detective uses power at random, cop does not use power: 18% chance detective is culted (1/10+1/11*9/10)
Detective uses power at random, cop uses power on known non-cult: 26% chance one of them is culted (1/10+2/11*9/10)
Detective uses power at random, cop uses power at random: 35% chance one of them is culted (1/10+1/9+2/11*(1-1/10-1/9)) ...Might have done that wrong
I'm assuming that cult can't redirect a power that is not being used, which is why using a power on someone you know is not cult is a little risky on odd nights.
I am more scared of mafia than cult, and getting info for a successful vig kill is important, so I would be against withholding the detective power.

