Published: 2017-04-04 18:16:42 BdST
Cook, who was appointed captain in 2012, led England in a record 59 tests including Ashes victories at home in 2013 and 2015 as well as series wins in India and South Africa.
The 32-year-old stepped down in February after a 4-0 defeat in the five-test series in India at the end of last year. He was replaced by Root, who will captain England for the first time against South Africa at Lord's on July 6.
"A lot will be made of it, there always is, it's a good story to write about but I think it'll be absolutely fine in terms of our relationship," Cook told British media.
The former captain said he hoped his transition back to just playing for the team would be smooth.
The left-handed opening batsman has scored more test centuries than any of his predecessors and is also England's most prolific run-getter with 11,057 runs in 140 tests.
With the emergence of a pair of batting prospects in India for England's fragile top order, Cook hoped he could continue making an impact with the willow.
Haseeb Hameed and Keaton Jennings became Cook's 10th and 11th opening partners since the retirement of Andrew Strauss, with both youngsters showing signs they might have bright futures after impressive debuts in India.
"You're always under pressure, that's what life is about," Cook said. "That is what playing international cricket or being a professional sportsman is. You're there to score runs.
"If you don't do that over a period of time, people will look elsewhere. That hasn't changed and that'll never change, whether it's myself or Jimmy Anderson, you've got to play to a certain level to be picked for England or even Essex."