# Solving \$x=|tau(x)|^{1/beta-1}(beta-1)^{1/beta}frac{pi/beta}{sin(pi/beta)}-|tau(x)|^{-1}+o(|tau(x)|^{-1})\$ for \$tau(x)\$

How do you solve the following equation for $$tau(x)$$?
$$begin{equation} x=|tau(x)|^{1/beta-1}(beta-1)^{1/beta}frac{pi/beta}{sin(pi/beta)}-|tau(x)|^{-1}+o(|tau(x)|^{-1}) end{equation}$$
The author of the paper (https://arxiv.org/pdf/1607.08794.pdf, page 13) states that
$$begin{equation} tau(x)=x^{-1}-b(beta)x^{-beta/(beta-1)}+o(x^{-1}),quad b(beta):=(beta-1)^{1/(beta-1)}left(frac{pi/beta}{sin(pi/beta)}right)^{beta/(beta-1)} end{equation}$$
is the solution, but I wonder how to get there.
For the function $$tau(x)$$ holds $$lim_{xrightarrow 0}=-infty, lim_{xrightarrow infty}=(beta-1)^{-1}$$ and $$tau(1)=0$$, $$beta>1$$.
Another user has helped me to understand that
$$begin{equation} tau(x)sim -b(beta)x^{-beta/(beta-1)} text{ for } xrightarrow0. end{equation}$$
(Approximating \$tau(x)\$ from \$x=|tau(x)|^{1/beta-1}(beta-1)^{1/beta}frac{pi/beta}{sin(pi/beta)}-|tau(x)|^{-1}+o(|tau(x)|^{-1})\$), but I’m not sure if that helps/how to apply that for this problem.
Any help is appreciated.