Kevin Pietersen belted 57, including seven fours and two sixes, before he holed out to mid on, while Ian Bell also fell in the final session when he edged James Muirhead (2-97) to Ben Rohrer at first slip for 35.
Joe Root (26) and Jonny Bairstow (11) survived until stumps and will resume the fight on Friday morning.
Josh Lalor (2-82) and Muirhead picked up two wickets each while Nic Bills (1-45) claimed the prize scalp of Cook (81).
England's quicks ensured their batters didn't have to spend too long in the field on Thursday morning after ripping through the hosts' tail inside the opening hour of play.
Steven Finn cashed in with three wickets to finish with flattering figures of 5-103, while Stuart Broad (4-37) and Boyd Rankin (1-63) each chipped in with a wicket as the CA XI lost 5-33.
Off-spinner Graeme Swann (0-66) did not bowl on day two as Cook allowed his seamers to run through the tail.
In reply, Cook and Trott (83) added 143 for the second wicket after losing Michael Carberry (four) early before both men were guilty of loose shots and edged to Peter Nevill behind the stumps.
Trott got off the mark with a boundary in an ominous sign of what was to come for the local bowlers.
The England number three raced to his half-century with his ninth four of the innings, a classy on drive, and added five more boundaries before giving his wicket away.
Trott was guilty of hanging his bat well away from his body and edged to Nevill who took a good diving catch off Lalor's bowling.
Cook continued to look in total control until he too played away from his body and edged to Nevill off Bills' bowling.
Earlier, Broad struck with the first ball of the morning when he drew a thin edge from Ryan Carters (94) to deny the third-string NSW wicketkeeper what would have been a well-deserved century.
Lalor fell three overs later, caught at second slip off Finn for a duck, before Nevill became Finn's fourth victim when he edged to Bairstow behind the stumps to depart for 83.
Rankin got his first wicket of the innings when Chris Tremain (one) slapped an easy catch to point before Finn sealed his five-wicket haul when Bills (six) was also caught by Carberry at point.
Alastair Cook, Jonathan Trott power England
