Question

Why did the United States become involved in a war that was difficult for it to win

Answer 1

Assuming your question is in reference to the Vietnam War, the United States got involved because of

domino theory and the policy of containment.

"Domino theory" was applied in justifying US involvement in the Vietnam War.  Domino theory said that the spread of communism in Vietnam would result in other countries in the region falling like dominoes to the influence of communism.  

Domino theory was a corollary to the overall policy of containment that developed as  the United States' policy regarding communism after the end of World War II.  George F. Kennan is known for recommending the policy of containment, which said that the best way to deal with the threat of communism was simply to try to keep it contained to the places where it already existed.