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"walk in a vain, important manner," Old English strutian "to stand out stiffly," from Proto-Germanic *strut- (cf. Danish strutte, German strotzen "to be puffed up, be swelled," German Strauß "fight"), from PIE root *ster- "strong, firm, stiff, rigid" (see sterile). Originally of the air or the attitude; modern sense, focused on the walk, first recorded 1510s. Cognate with Old English ðrutung "anger, arrogance" (see throat). To strut (one's) stuff is black slang, first recorded 1926, from strut as the name of a dance popular from c.1900.
To display one's virtuosity, esp in a saucy, provocative way
[1926+ Black; fr dances featuring a strut like the turkey cock's, popular from around 1900]