I would like to find a two-term or a three-term asymptotic expansion of $x_{n}$ the unique solution of $$x_{n}=\frac{1}{\tan(x_{n})}$$ on the interval $]n\pi,n\pi+\pi[ $
We have: $$ x_{n}=n\pi+\arctan(\frac{1}{x_{n}})$$
So $$x_{n} \sim_{n\rightarrow \infty} n\pi$$
What is the method to find the next terms of the asymptotic expansion?