Question

How did the United States show its dominance over the Philippines

Answer 1

through military action
by political measures
with economic pressure
through diplomatic relations

Answer 2

Mainly military action.

In short, during the Spanish American War, the Philippines declared their independence from Spain and named itself a Republic. After the War, however, the United States were given the islands in exchange for cash as a part of the treaty negotiations with Spain.

The problem was that the US did not recognize them as a separate nation, and when the Philippines attempted to assert its independence, it was put down by US military forces in the Philippine-American War.