The middle name probably indicates that his mother's maiden name was Cook.
John Cook Pettit , son of John Pettit of Abbots Hall, Essex, gentleman, was an apprentice of Charles Hougham. The date of 30th May 1781 is given in the extracts of apprentice records on Britishorigins.com followed by [free] Curriers Company . This probably means that Cook was free of the Curriers Company on that date.
1790 National Archive record Chancery Record C 111/207
GLEIG v BEATY: Lease of Thomas Beaty to John Cook Pettit of Whitefriars, goldsmith, of two houses in Whitefriars, London
3rd March 1792 London Gazettte 13394 announced him Bankrupt. Described as Goldsmith, Bucklemaker, Dealer and Chapman.
11th April 1792- The Times advertised the sale of his furniture and effects, including two brick forges, bellows and ironwork etc.