The words trooper, trouper sound the same but have different meanings and spellings. Why do trooper, trouper sound the same even though they are completely different words?
The answer is simple: trooper, trouper are homophones of the English language.
A member of a unit of cavalry.
A cavalry horse.
A mounted police officer.
A state police officer.
A member of a theatrical company.
A veteran actor or performer.
A reliable, uncomplaining, often hard-working person.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").