In a careful, tactful, and subtle way; avoiding drawing attention or revealing secrets.
From 'discreet' (careful, prudent) plus the adverbial suffix '-ly.' The root comes from Latin 'discretus' (separated, distinct), emphasizing keeping things separate and careful.
People constantly confuse 'discretely' with 'discretly'—but 'discretely' relates to 'discrete' (separate/distinct) while 'discreetly' relates to 'discreet' (tactful). This spelling split perfectly shows how English preserves two meanings from the same Latin root!
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