if I try the cayenne pepper diet and cheated 1 day will it still work by the end of the 10 days?

Considering the other days are close to 0 calories, then yes.

The Master Cleanse works because it's basically water fasting and you're eating 0 calories.

I would say it depends on how many days you have completed on the diet itself. If you had succeeded 9 days, then yes you will still benefit from the 9 days. If this is your 2nd day, maybe you don’t benefit as much.