In a way that shows you are completely sure about something, even when you shouldn't be; with arrogant confidence.
From 'cocksurel' (14th century English), combining 'cock' (rooster, symbol of arrogance) and 'sure,' suggesting the overconfident strut of a rooster. The adverbial form adds '-ly' to create the adverbial modifier.
The word 'cock' in this phrase has nothing to do with roosters being right—it's all about their attitude! Roosters puff out their chests and crow loudly even when they're about to walk into danger, which is exactly what arrogant people do.
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