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Old English tæppe "narrow strip of cloth used for tying, measuring, etc.," of uncertain origin, perhaps a back-formation from Latin tapete "carpet." The original short vowel became long in Middle English.
Tape recorder "device for recording sound on magnetic tape" first attested 1932; from earlier meaning "device for recording data on ticker tape" (1892), from tape in the sense of "paper strip of a printer" (1884). Tape-measure is attested from 1873; tape-delay is from 1968.
c.1600, from tape (n.); meaning "to make a tape recording" is from 1950. Related: Taped; taping.
To improvise, tergiversate, etc, in order to hide one's ignorance: I didn't read the poop sheet, so I had to tap dance when the question came (1970s+ Army)