Used when talking about losing something, often weight, through physical activity or effort. I'll have to sweat off this chocolate cake at the gym.
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Colloquial Tone
'Sweat off' is casual language, suitable for conversing with friends, not formal situations. You've worked hard today, time to go sweat off the day's stress.
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Metaphoric Usage
Sometimes used metaphorically to mean working hard on non-physical tasks. I need to sweat off this debt by picking up extra shifts.