Some Like It Hot

It is February 1929 in the city of Chicago, during the era of prohibition. Joe (Tony Curtis) is an irresponsible jazz saxophone player, gambler and ladies' man; his friend Jerry (Jack Lemmon) is a sensible jazz double-bass player; both are working in a speakeasy (disguised as a funeral home) owned by mob gangster "Spats" Colombo (George Raft). When the joint is raided by the police after being tipped off by informant "Toothpick" Charlie (George E. Stone), Joe and Jerry fleeâonly to accidentally witness Spats and his henchmen exacting his revenge on "Toothpick" and his own gang (inspired by the real-life Saint Valentine's Day Massacre). Penniless and in a mad rush to get out of town, the two musicians take a job with Sweet Sue (Joan Shawlee) and her Society Syncopators, an all-female band headed to Miami. Disguised as women and renaming themselves Josephine and Daphne, they board a train with the band and their male manager, Bienstock. Before they board the train, Joe and Jerry notice Sugar Kane (Marilyn Monroe), the band's vocalist and ukulele player.
Joe and Jerry become enamored of Sugar and compete for her affection while maintaining their disguises. Sugar confides that she has sworn off male saxophone players, who have stolen her heart in the past and left her with "the fuzzy end of the lollipop". She has set her sights on finding a sweet, bespectacled millionaire in Florida. During the forbidden drinking and partying on the train, Josephine and Daphne become intimate friends with Sugar, and have to struggle to remember that they are supposed to be girls and cannot make a pass at her.
Once in Miami, Joe woos Sugar by assuming a second disguise as a millionaire named Junior, the heir to Shell Oil, while feigning disinterest in Sugar. An actual millionaire, the much-married aging mama's boy Osgood Fielding III, (Joe E. Brown) tries repeatedly to pick up Daphne, who rebuffs him. Osgood invites Daphne for a champagne supper on his yacht. Joe convinces Daphne to keep Osgood occupied onshore so that Junior can take Sugar to Osgood's yacht, passing it off as his. Once on the yacht, Junior explains to Sugar that, due to psychological trauma, he is impotent and frigid, but that he would marry anyone who could change that. Sugar tries to arouse some sexual response in Junior, and begins to succeed. Meanwhile, Daphne and Osgood dance the tango ("La Cumparsita") till dawn. When Joe and Jerry get back to the hotel, Jerry explains that Osgood has proposed marriage to Daphne and that he, as Daphne, has accepted, anticipating an instant divorce and huge cash settlement when his ruse is revealed. Joe convinces Jerry that he cannot actually marry Osgood.
The hotel hosts a conference for "Friends of Italian Opera", which is in fact a front for a major meeting of various chapters of the mafiosi. Spats and his gang from Chicago recognize Joe and Jerry as the witnesses to the Valentine's Day murders. Joe and Jerry, fearing for their lives, realize they must quit the band and leave the hotel. Joe breaks Sugar's heart by telling her that he, Junior, has to marry a woman of his father's choosing and move to Venezuela. After several chases, Joe and Jerry witness additional mob killings, this time of Spats and his crew. Joe, dressed as Josephine, sees Sugar onstage singing that she will never love again. He kisses her before he leaves, and Sugar realizes that Joe is both Josephine and Junior.
Sugar runs from the stage at the end of her performance and manages to jump into the launch from Osgood's yacht just as it is leaving the dock with Joe, Jerry, and Osgood. Joe tells Sugar that he is not good enough for her, that she would be getting the "fuzzy end of the lollipop" yet again, but Sugar wants him anyway. Jerry, for his part, comes up with a list of reasons why he and Osgood cannot get married, ranging from a smoking habit to infertility. Osgood dismisses them all; he loves Daphne and is determined to go through with the marriage. Exasperated, Jerry removes his wig and shouts, "I'm a man!" Osgood simply responds, "Well, nobody's perfect."

Question: Who woos sugar?
Answer: Joe