Old English swætan "perspire, work hard," from the source of sweat (n.). Meaning "to be worried, vexed" is recorded from c.1400. Related: Sweated; sweating. Colloquial no sweat "no problem" attested from 1963.
Old English swat "sweat," which became Middle English swote, but altered under the influence of the verb, from Proto-Germanic *swaita (cf. Old Saxon, Old Frisian swet, Old Norse sveiti, Danish sved "sweat," Swedish svett, Middle Dutch sweet, Dutch zweet, Old High German sweiz, German Schweiß), from PIE *sweid-/*swoid- (cf. Sanskrit svedah "sweat," Avestan xvaeda- "sweat," Greek hidros "sweat, perspiration," Latin sudor, Lettish swiedri, Welsh chwys "sweat"). Sweat equity is from 1968.