Question 25
There is a unique sequence of integers $a_1,$ $a_2,$ $\cdots a_{2023}$ such that $$
\begin{gathered} \tan 2023x\\ {}=\tiny\dfrac{a_1\tan x+a_3\tan^3 x+a_5\tan^5 x+\cdots+a_{2023}\tan^{2023}x}{1+a_2\tan^2 x+a_4\tan^4 x\cdots+a_{2022}\tan^{2022}x} \end{gathered}
$$ whenever $\tan 2023x$ is defined. What is $a_{2023}?$
\begin{gathered} \tan 2023x\\ {}=\tiny\dfrac{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} \end{gathered}
$$ whenever $\tan 2023x$ is defined. What is $a_{2023}?$