There is a unique sequence of integers $a_1,\dots,a_{2025}$ such that $\tan2025x=\dfrac{a_1\tan x+a_3\tan^3x+\cdots+a_{2025}\tan^{2025}x}{1+a_2\tan^2x+\cdots+a_{2024}\tan^{2024}x}$ whenever $\tan2025x$ is defined. What is $a_{2025}$?

1. From $\tan(nx)=\mathrm{Im}[(1+i\tan x)^n]/\mathrm{Re}[(1+i\tan x)^n]$, the top coefficient (of $\tan^nx$) is $(-1)^{(n-1)/2}\binom nn$.
2. With $n=2025$: $(2025-1)/2=1012$, which is even, so $(-1)^{1012}=1$.
3. $a_{2025}=\boxed1$.