Cook, who got a reprieve on 15 when Ravindra Jadeja dropped a sitter at third slip off debutant Pankaj Singh's bowling, battled hard to reach 82 off 188 balls and shared a 131 run unbroken stand for the second wicket with in-form Gary Ballance.
Ballance scored his third successive 50 plus score in as many Tests to reach 72 at tea off 131 balls.
Opting to bat on a flat Ageas Bowl track, Cook overcame his poor run of form to hit his first half-century in 10 innings.
Although it wasn't conducive for quality pace bowling, the Indian attack in the absence of an injured Ishant Sharma lacked discipline.
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Mohammad Shami (1-41) was the lone wicket-taker in two sessions so far having removed Sam Robson (26) after a 55 run opening stand.
Bhuvneshwar Kumar (0-50), Pankaj Singh (0-41), Ravindra Jadeja (0-24), Rohit Sharma (0-21) and Shikhar Dhawan (0-3) were the other bowlers deployed.
Debutant Pankaj may consider himself unlucky not to have got Cook's wicket but bowled better compared to his other two teammates.
The odd ball did something, but overall the bowling didn't look sharp enough in the absence of genuine aid. And so the two batsmen carried on in the same vein they had before the break, enhancing their 23-run stand.