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 to be 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 said, adding that he hoped his transition back to just playing for the team would be smooth:
I feel like I can get back in the troops and get on with it and hopefully score some runs and help England win cricket games, which ultimately is what everyone wants to see.
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.
Read the full article on eurosport.co.uk: Relationship with Root will be absolutely fine, says Cook