What is hypocrisy? Is social appeasement—such as showing courtesy and avoiding embarrassment with a person whose actions you dislike and about whom negative rumors circulate—considered hypocrisy? What is the correct Shar'i (Islamic legal) conduct in such cases?

Your courtesy towards a woman who displays herself immodestly (mutabarrijah), by smiling and being gentle with her, out of fear of her evil and harm, while you detest her state and do not assist her in her sin or approve of it—this is not hypocrisy, but rather a form of conciliation (mudārah). Evidence for its permissibility is found in what was narrated in the Two Sahihs from Aisha, may Allah be pleased with her, that the Prophet, peace and blessings be upon him, said about a man: "What a bad brother of his tribe!" But when the man entered, the Prophet, peace and blessings be upon him, brightened his face towards him and was at ease with him. He then said: "Indeed, the worst of people in station before Allah on the Day of Resurrection is he whom people abandon to avoid his evil."