OP have you ever read up about teenagers? There's good books out there.
That aside, I'm not sure I'd like to live with you either, given the 'catching him out' dynamic you've created. What competition exactly are you trying to 'win', because it sounds pretty fucked up to me that you're feeling that way and not acknowledging it as the resentment that it is.
My DS is home from university. If I want him to feed the cats, I say, 'Please will you feed the cats? That'd be helpful.'
My DP had teenagers living with him full-time when we met, for complex reasons. I wouldn't have dreamed of moving in with them (or marrying him). We are having to play the long game. I know it's not the answer you're looking for, because you're there now. So I'd advise giving the boy some space, stop trying to catch him out, and get some personal counselling about your feelings of resentment. Allow your DP to talk about his son and listen.
I don't feel the same way about my DP's DC as my own. But I'm fond of them, I help them, and I work at communicating with them. If one of them lies about something small I either ignore it, or try to find out (gently) what the problem might be. Often it just means 'back off' because they don't want to say to dad's girlfriend, 'What the fuck has it got to do with you?' But the main thing is I talk to their dad about it, privately, without blame or recrimination. Otherwise, what's the point of it all - what's the point of the relationship if everyone's unhappy?