Well, here we are now with word that you are able to use Chromecast on a tab which you would much rather prefer to have it remain a secret between you and your browser, so that it will not show up in the browser history sometime down the road. Since in Incognito Mode, extensions are turned off by default, but thankfully, flipping the switch is a snap and you are able to do so for individual extensions.
All you need to do is type chrome://extensions and you will be sent to the Extensions page. Over there, do scroll down until you find “Google Cast,” before you click the “Allow in incognito” checkbox. This would allow it to function in Incognito mode, letting you Cast whatever you want without having others see what you do not want them to check out in the first place.