trooper vs Rough Rider

trooper and Rough Rider both are nouns.

Word	Noun	Adjective	Verb	Adverb
trooper	Yes	No	No	No
Rough Rider	Yes	No	No	No

As nouns, Rough Rider is a hyponym of trooper; that is, Rough Rider is a word with a more specific, narrower meaning than trooper:

trooper: a soldier mounted on horseback
Rough Rider: a member of the volunteer cavalry regiment led by Theodore Roosevelt in the Spanish-American War (1898)

Other hyponyms of trooper include cuirassier, dragoon, hussar, lancer.

trooper (noun)	Rough Rider (noun)
a mounted police officer	a member of the volunteer cavalry regiment led by Theodore Roosevelt in the Spanish-American War (1898)
a soldier in a motorized army unit
a state police officer
a soldier mounted on horseback