Well the question describes the problem.
I know that the YouTube app is not using secure connections, and I want to force the YouTube app to use secure connections. Can this be done?
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on your phone. You can install firefox if you not already have it. Then open firefox and install this extension named https everywhere. it forces every site to load in https if it has https capability. It includes youtube in its list, so it may solve your problem for using youtube in https mode on your android.