Final answer:
The Cold War during Eisenhower's presidency evolved through the policy of brinkmanship and the containment of communism in developing nations such as Vietnam.
Step-by-step explanation:
The Cold War during Eisenhower's presidency evolved in several ways. Firstly, Eisenhower adopted a policy called brinkmanship, which involved a willingness to go to the brink of war to maintain peace. This was demonstrated by the United States' response to events such as the Hungarian uprising and the Suez Crisis. Additionally, Eisenhower's administration focused on containing communism and preventing its spread by supporting anti-communist governments in developing nations. This was exemplified by the United States' involvement in Vietnam, where they initially supported France and later replaced them in the fight against communist forces led by Chi Minh.