not exactly.the double angle formula for cos states that cos2x = 2cos^2x - 1 . You cannot "cancel" away cos on it's own. It is always accompanied by a something(in this case,x)
and also for proving, you are only allowed to algebraically manipulate one side such that it looks like the other. cross multiplication and things like that are not allowed because if you do that it means that it is an equation and not an identity(correct me if I am wrong here)