What is it called when you believe in something and speak out against it?

The term you're looking for is "cognitive dissonance." Cognitive dissonance is the mental discomfort that occurs when you hold two or more contradictory beliefs, ideas, or values. For example, if you believe that smoking is harmful to your health, but you continue to smoke, you are experiencing cognitive dissonance. This discomfort can lead to a variety of negative emotions, such as guilt, shame, or anxiety. In order to reduce cognitive dissonance, people often change one or more of their beliefs or behaviors. For example, you may decide to quit smoking or you may try to rationalize your behavior by telling yourself that the risks of smoking are exaggerated.