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?