Proof of the reconcilation of the geometric form of cross product with the algebraic form.

Question

In Arfken's "Mathematical Methods for physicists" he stated that:

$(A\times B)\cdot(A\times B) = A^2B^2-(A\cdot B)^2=A^2B^2-A^2B^2\cos^2(\theta)$

How did he arrived to that? He said that he is going to expand C (The vector product of A & B ) in Component form.
any Detailed solution will be helpful