How to show the following determinant inequality problem

Question

Let $A\in M_2(\Bbb{R})$ be such that $\det\,A\geq0$. Show that $$\det(A^n+I_2)+\det(A^n-I_2)\geq\frac{1}{2^{n-2}}\left(\det\,A+1\right)^n$$

I have shown that $$\det(A^n+I_2)+\det(A^n-I_2)\geq2((\det A)^n+1)$$ From this how I can proceed?