"Brinkmanship" is a political strategy. It is when you try to achieve an outcome by pushing dangerous situations to create a bigger conflict: for example, creating many little fights to create a war. "To be exposed" is a phrase that means someone's secret intentions, or true character, were found out. So, in this text, Syria's political strategy of creating dangerous conflicts showed who Nasser really was. In this text, it seems to imply he wasn't a good political leader. I hope that helps!

1) The "-manship" ending means ability or skill in a field. For example "seamanship" is the art of managing ships well. One could also talk about horsemanship, craftsmanship, marksmanship, swordsmanship, and so forth. 2) In 1947 a writer named Stephen Potter wrote a humorous and satirical book entitled "gamesmanship." It didn't mean skill at playing some game fairly. It meant skill at winning games generally, any game, by doing unsporting things. It meant winning games, not by being better at the game itself, but by being better at the art of playing games. For example, a gamesmanship technique is to get your opponent angry so that they will lose their concentration and make mistakes. It works whether you are playing golf or chess. 3) In the 1950s, US Secretary of State John Foster Dulles had an approach of going "to the brink of war," as close to war as possible, to frighten the Soviet Union into backing down. That is, by bluffing--making your adversary think you are ready to go further than you really are. Critics began to call this "brinksmanship" or "brinkmanship," by analogy with "gamesmanship." 4) "Brinksmanship" gradually became an accepted word, no longer connected with Dulles, for this kind of bluffing.