1620s (implied in insincerely), from Latin insincerus "spoiled, corrupted; not genuine, not pure, adulterated," from in- "not" (see in- (1)) + sincerus "genuine, candid" (see sincere). Related: Insincerely.
word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."
In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
1530s, "pure, unmixed," from French sincere (16c.), from Latin sincerus, of things, "whole, clean, pure, uninjured, unmixed," figuratively "sound, genuine, pure, true, candid, truthful," of uncertain origin. The ground sense seems to be "that which is not falsified." Meaning "free from pretense or falsehood" in English is from 1530s.
There has been a temptation to see the first element as Latin sine "without." But there is no etymological justification for the common story that the word means "without wax" (*sin cerae), which is dismissed out of hand by OED and others, and the stories invented to justify that folk etymology are even less plausible. Watkins has it as originally "of one growth" (i.e. "not hybrid, unmixed"), from PIE *sm-ke-ro-, from *sem- "one" (see same) + root of crescere "to grow" (from PIE root *ker- (2) "to grow"). De Vaan finds plausible a source in a lost adjective *caerus "whole, intact," from a PIE root meaning "whole."