by 1942, noun ("a cowardly escape, an evasion") and verb ("sneak off, escape, give up without trying"), American English slang, perhaps from cop a plea (c. 1925) "plead guilty to lesser charges," which is probably from northern British slang cop "to catch" (a scolding, etc.); as in cop a feel "grope someone" (1930s); see cop (v.). Sense of "evade an issue or problem" is from 1960s.
also cutout, 1851, in reference to a kind of switch on a circuit to cut out an instrument, from the verbal phrase, from cut (v.) + out (adv.). The verbal phrase is attested from c. 1400 as "cut so as to take out;" from 1550s as "fashion or shape by cutting;" from 1736 as "remove, excise, omit." From 1640s as "be naturally formed or fashioned" (for some specified purpose).