Cook was unbeaten on 167 (323b, 22x4, 1x6) after being dropped at 156 by Ishant Sharma who grassed a sitter, while Trott returned to form giving the skipper fine support with an unbeaten 66 (172 balls, 8x4).
England now trail India's first innings total by just 20 runs and are firmly in the driver's seat.
Resuming the day at 216 for one, England did not have any problem against Indian bowlers who looked listless while the fielding was poor.
The Indians were made to suffer as the duo put on an undefeated 131-run for the second wicket.
Cook had one edgy moment in the morning when he missed a sweep off Ashwin as the ball struck the pad outside the line inviting a huge appeal, but thereafter the English skipper continued from where he left yesterday.
The duo of Ashwin and Ojha continued bowling for 15 overs together before India took the new ball in the 88th over but the result was no different.
Cook had no trouble negotiating the home bowlers as he notched up his seventh Test 150, clobbering Zaheer through mid-wicket as he continued his record-breaking innings.