In an arrogant manner; with excessive pride or self-importance.
From 'arrogant' plus the adverbial suffix '-ly,' which transforms adjectives into adverbs describing how an action is performed. Standard English word formation.
The difference between 'arrogant' (the attitude) and 'arrogantly' (the action) matters—you can be arrogant and quiet, but when you do something 'arrogantly,' everyone knows it, which is probably why people use the adverb as a genuine insult.
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