Determinant with trigonometric functions 
If
  $$
\begin{vmatrix}
\sin 2x   & \cos^2x &  \cos 4x \\
\cos^2x & \cos2x  & \sin^2x \\
\cos^4x & \sin^2x & \sin 2x \\
\end{vmatrix} = a_0 + a_1\sin x + a_2\sin^2x +\cdots+ a_n \sin^n x
$$
Then what is the value of $a_0$?

How do I solve this?
Thank you so much!!

As noted in the comments, let $x=0$. Then we have
$$
\begin{vmatrix}
\sin 2x   & \cos^2x &  \cos 4x \\
\cos^2x & \cos2x  & \sin^2x \\
\cos^4x & \sin^2x & \sin 2x \\
\end{vmatrix} =  a_0 + a_1\sin x + a_2\sin^2x +\cdots+ a_n \sin^n x \\
\iff
\begin{vmatrix}
0   & 1 &  1 \\
1 & 1  & 0 \\
1 & 0 & 0 \\
\end{vmatrix} = a_0 + a_1 \cdot 0 + a_2\cdot 0 +\cdots+ a_n \cdot 0 = a_0
$$
Can you take it from here?
