Let $G_1,G_2\in\mathbb{C}$, such that ${\rm Re~}G_1G_2^*>0$. Let $a,\theta>0$. Assume $\cos\theta\neq0$ and $\sin\theta\neq0$. I need to show that for sufficiently small $a$, \begin{equation} \prod_{k=0}^\infty\left(I_2-a\begin{bmatrix}(\cos k\theta)^2&\cos k\theta\sin k\theta\\\cos k\theta\sin k\theta&(\sin k\theta)^2\end{bmatrix}\begin{bmatrix}{\rm Re~}G_1G_2^*&{\rm Im~}G_1G_2^*\\{\rm -Im~}G_1G_2^*&{\rm Re~}G_1G_2^*\end{bmatrix}\right)=0. \end{equation}
I figured out that the condition ${\rm Re~}G_1G_2^*>0$ is equivalent to $|\angle(G_1/G_2)|<\pi/2$ (that is, $G_1$ is within 90$^\circ$ of $G_2$ in complex plane), but I have no idea how to proceed. Any idea or hint is appreciated. Thanks
Note: $I_2$ denotes identity matrix of dimension 2 and $^*$ denotes the complex conjugate transpose