AMC 12 2023a Problem 25

There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that $$\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}$$whenever $\tan 2023x$ is defined. What is $a_{2023}?$

$\textbf{(A) } -2023 \textbf{(B) } -2022 \textbf{(C) } -1 \textbf{(D) } 1 \textbf{(E) } 2023$