【Cool】 , 【composed】 , 【collected】 , 【unruffled】 , 【imperturbable】 ,  【unflappable】 , 【nonchalant】 are comparable when applied to persons, their manners, appearance, temper, or acts, in the sense of showing or seeming to show freedom from agitation or excitement.

【Cool】 (see also COLD) basically implies such self-control that no hint is given of any emotion or motive that might warm, inflame, excite, or impassion. Specifically, it may further imply detachment or dispassionateness or calm courage in assault or under attack or deliberateness or determination in gaining one's ends or calm assurance or effrontery or actual or seeming indifference.

【Composed】 implies the freedom from signs of agitation or excitement that is characteristic of a decorous sedate temperament or is the result of self-discipline.

【Collected】 stresses a concentration of the mind or spirit with resulting elimination of all distractions; otherwise it differs little from 【composed】 .

【Unruffled】 implies coolness, placidity, and often, poise, in the midst of excitement or when there is cause for agitation.

【Imperturbable】 implies such coolness and assurance that one cannot be abashed, annoyed, disconcerted, alarmed, or otherwise disturbed; it usually implies a temperamental rather than an acquired frame of mind.

【Unflappable】 is a somewhat informal synonym of 【imperturbable】 , and like the latter stresses ability to resist what tends to disturb.

【Nonchalant】 stresses an easy coolness of manner or casualness that suggests indifference or unconcern; it often connotes lightheartedness or offhandedness.