Police investigations are continuing.
The death has been referred to the coroner who will determine the cause of death.
Mr Cossey's family being supported by police and victim support.
A police spokeswoman said Canterbury Police Search and Rescue and members from Alpine Cliff Rescue recovered the body about 8.30pm yesterday from a remote and technically difficult location in the hills.
Low cloud and inclement weather prevented a helicopter being used.
Aucklander Mr Cossey had arrived in Christchurch on December 4 to work at Mercer Stainless.
On the morning of December 5, the qualified engineer and stock-car racing enthusiast had arrived at work in the morning but left after just 20 minutes left with authorisation.
He was seen on CCTV footage at Eastgate mall in Linwood about 1pm but wasn't seen again.
His brother Leigh and father Geoff came down from Auckland to help in the search.
They visited local shopping malls and handed out information leaflets, hoping for any clues to his out-of-character disappearance.
A large scale search on ground, water and from the air had been underway when a tramper came across his body yesterday.
Today police explained the "rigorous process" they follow in order to formally identify a body and ensure that next of kin are fully informed.
"The identify of a deceased person is only confirmed once both of these processes are completed," a statement said.
"Formal identification in this case was completed at mid-morning today.
"Canterbury Police extend their condolences to Mr Cossey's family and would like to thank the volunteers and public who have helped in the search for him since he was reported missing."
