I have this function for reals $0<x<\pi$. $$f(x)=(3 \pi -2 x)^2 \sin \left(\frac{1}{2} \left(\csc ^{-1}\left(\frac{4 \pi (\pi -x) \csc \left(\frac{\pi ^2}{\pi -x}\right)}{4 x^2-8 \pi x+5 \pi ^2}\right)-\frac{\pi x}{\pi -x}+\pi \right)\right)\\-(\pi -2 x)^2 \sin \left(\frac{1}{2} \left(\csc ^{-1}\left(\frac{4 \pi (\pi -x) \csc \left(\frac{\pi ^2}{\pi -x}\right)}{4 x^2-8 \pi x+5 \pi ^2}\right)+\frac{\pi ^2}{\pi -x}\right)\right)$$
By the code below, we are sure that this function has infinitely many roots which are rational multiples of $\pi$.
I try NSolve to obtain all the roots of the function over the domain, but, it seems that Mathematica needs much time to produce all the roots.
Is there any alternative for NSolve or Solve to obtain all the roots? Any comments are appreciated.
f = -(\[Pi] - 2 x)^2 Sin[ 1/2 (\[Pi]^2/(\[Pi] - x) + ArcCsc[(4 \[Pi] (\[Pi] - x) Csc[\[Pi]^2/(\[Pi] - x)])/( 5 \[Pi]^2 - 8 \[Pi] x + 4 x^2)])] + (3 \[Pi] - 2 x)^2 Sin[ 1/2 (\[Pi] - (\[Pi] x)/(\[Pi] - x) + ArcCsc[(4 \[Pi] (\[Pi] - x) Csc[\[Pi]^2/(\[Pi] - x)])/( 5 \[Pi]^2 - 8 \[Pi] x + 4x^2)])];
x/.NSolve[f==0 && 0<x<\[Pi]]