SINGAPORE - U.S. President Donald Trump and North Korean leader Kim Jong Un shook hands and smiled as they started a historic summit on Tuesday, just months after they traded insults and threats of nuclear war.For both men, the summit, the first between leaders of North Korea and the United States, is likely to be a defining moment of their careers.The meeting would have been unthinkable last year when tensions spiraled in the region over North Korea's nuclear and missile programs as it raced toward the goal of developing a nuclear-tipped missile capable of hitting the United States.Trump vowed to prevent that happening and threatened North Korea with "fire and fury like the world has never seen." He mocked Kim as "little rocket man" on a suicide mission.Kim denounced the U.S. president as the "mentally deranged U.S. dotard" who would be "tamed ... with fire."But once North Korea conducted an underground test of a thermonuclear bomb last September, Trump began taking Kim more seriously.Trump can look back to President Richard Nixon's 1972 trip to China, when the ardent anti-communist defied conventional wisdom to meet Chairman Mao Zedong, irrevocably shifting the Cold War balance and setting China on a path to opening up.Trump and his senior officials have in recent weeks been referring to Kim as "Chairman."For Trump, cutting a deal to end the North Korean nuclear threat with his approach to Kim, in defiance of the U.S. security establishment's long-held ways of dealing with the North, would be a success unmatched by any predecessors.It would justify his supporters' faith in his art of deal-making and allow him to repudiate the naysayers with a foreign-policy success he will be able to hail. Some supporters have said he could win the Nobel Peace Prize.For Kim, about half Trump's age and representing the third generation of North Korea's ruling dynasty, the summit affords both him and his long-isolated country the international legitimacy his father and grandfather could only dream of.North Korea has faced years of economic sanctions over its nuclear and missile programs since it conducted its first nuclear test in 2006.