An actor called Jay "Slim" Talbot often stood in for Gary Cooper, as they looked very similar, but this is not him.
The man in white in the puzzle picture is John Duff, who is considered to be the first of the "Bentley Boys".
He was born in China of Canadian parents, but came to England and joined the army at the outbreak of WWI. He was injured at the Battle of Passchendaele, and after the war set up a car dealership, later selling Bentleys.
He raced Bentleys at Brooklands, and made a private entry in the 1923 Le Mans 24-hour race partnered by Frank Clement, who was a Bentley employee. Apparently W.O. Bentley was not keen on the idea, but eventually agreed to have the car prepared by the factory. The car finished fourth, but in the following year Duff and Clement won the race.
Below is the 1924 Le Mans winning 3-litre Bentley. Duff is on the right, with Clement (left) and Walter Bentley.