No Exclusive Authority To Declare War
Despite the fact that the president must sign a declaration of war, the president does not have exclusive authority to declare it. War can only be started by a vote of Congress. At least 11 times, Congress has declared war on another country, the first of which was against the United Kingdom in 1812.
World War II was the last time the United States Congress declared war on another nation. Rather than declaring war, the United States has only been involved since then in situations that necessitated the use of force.