How can you show that $$ \det(A + B^T) = \det(A^T + B)$$ for any $n\times n$ matrices $A$ and $B.$
Prove that $$(A+B)^T=A^T+B^T$$ and that
$$\det A^T=\det A$$
and you're done. For the second one, consider expanding the determinant "column-wise" in one case, and "row wise" in the other. In fact, you can use the cofactor expansion of the determinant to prove the claim by induction.