If someone is a trouper he or she does what needs to be done without complaining or whining. A trouper is also part of a troupe, or a group of people, usually an acting troupe or theatre troupe.
If someone is a trooper he or she is a soldier at entry level or an officer in the police. In British English it is also a ship used to move troops.
Both words come from the French troupe, which carries the same meaning as today.
Some dictionaries list trooper as a synonym of trouper, however, this seems to be by association and misuse, rather than a valid connection.
Examples
A trouper to the last, Lynda launched the book by putting on a show-stopping display of bravery and — immaculately dressed and made-up — facing the world with immense courage. [The Daily Mail]
Tonight, he will play two more shows at the Target Arena, five hours of the kind of all-out, sweaty, soul-baring, passionate performance that could make even the most hardened showbiz trouper weak at the knees. [Irish Independent]
Delaware State Police say a male suspect has died following a trooper-involved shooting in Claymont, Delaware. [CBS Philadelphia]
State police spent about $11 million on the manhunt for a survivalist charged in the ambush slaying of a trooper outside his barracks, according to a cost breakdown released Friday. [ABC News]
There’s no pub in Halls Gap…but there is a zoo. And what a little trouper it is, boasting 160 species of animal such as the central netted dragon, spinifex hopping-mouse, veiled chameleon, pygmy marmoset (all true), and more “pattable” types like domestic pony donkey, elk, giraffe and exotics like Himalayan tahr. [Herald Sun]