Your first mistake is in the very first line: What is ##\vec{A}\vec{B}\cos(\theta+\phi)## supposed to mean? You can't just stick two vectors next to each other like that. ##\vec{A}\cdot\vec{B}## makes sense because it's saying your dotting the two vectors together; similarly, ##\vec{A}\times\vec{B}## makes sense because it's saying you're taking the cross product. ##\vec{A}\vec{B}## doesn't mean anything.
Your second mistake is that you're using the very identity you're trying to prove. So far, you're using the fact that ##\vec{A}\cdot\vec{B} = |\vec{A}||\vec{B}|\cos(\theta+\phi)##. You need to find a second way to express the dot product.