If you have done something wrong and you brazen it out, you behave confidently in order not to appear ashamed, even though you probably do feel ashamed.
If you are caught simply argue that 'everyone does it' and brazen it out. [V it P]
The president brazened out his misdeeds. [VERB PARTICLE noun]
He would brazen the matter out and he would do so in the most robust manner possible. [VERB noun PARTICLE]