Question

how did americans find out they were lied to about the operations in Vietnam

Answer 1

Confidential papers from the pentagon leaked out to the public and the Americans found out that they were lied to about the operations in the Vietnam War.

Answer 2

Generally by talking to their sons, fathers, brothers or cousins when the latter came home from the war. Those who focus on the role of the media seem often to forget that nearly 1 in 5 American men of the era actually served in Vietnam at one time or another, and if we add in their family and friends, it's obvious far more people got their information about Vietnam from a family member who'd actually served in contry than ever watched Walter Cronkie or read the New York Times.