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I have this function for $0<x<\pi$. $$f(x)=(3 \pi -2 x)^2 \sin \left(\frac{1}{2} \left(\csc ^{-1}\left(\frac{4 \pi (\pi -x) \csc \left(\frac{\pi ^2}{\pi -x}\right)}{4 x^2-8 \pi x+5 \pi ^2}\right)-\frac{\pi x}{\pi -x}+\pi \right)\right)\\-(\pi -2 x)^2 \sin \left(\frac{1}{2} \left(\csc ^{-1}\left(\frac{4 \pi (\pi -x) \csc \left(\frac{\pi ^2}{\pi -x}\right)}{4 x^2-8 \pi x+5 \pi ^2}\right)+\frac{\pi ^2}{\pi -x}\right)\right)$$

By the code below, we are sure that this function has infinitely many roots which are rational multiples of $\pi$.

My question

How can I make sure if this function also has any other roots which are not rational multiples of $\pi$? I mean can I claim that all the roots of this function are rational multiples of $\pi$?

I tried NSolve to obtain all the roots and then check whether they are rational multiples of $\pi$ or not, but, it seems that Mathematica needs much time to produce 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)])];

f2[y_] = f /. x -> y Pi // Simplify[#, y \[Element] Rationals] &;

sols = Cases[  Flatten[Table[{a/b, f2[a/b]}, {a, 1, 200}, {b, 1, 200}], 1], {_,  0}] // Quiet //Union

(*{{1/2, 0}, {3/4, 0}, {5/6, 0}, {7/8, 0}, {9/10, 0}, {11/12, 0}, {13/
  14, 0}, {15/16, 0}, {17/18, 0}, {19/20, 0}, {21/22, 0}, {23/24, 
  0}, {25/26, 0}, {27/28, 0}, {29/30, 0}, {31/32, 0}, {33/34, 0}, {35/
  36, 0}, {37/38, 0}, {39/40, 0}, {41/42, 0}, {43/44, 0}, {45/46, 
  0}, {47/48, 0}, {49/50, 0}, {51/52, 0}, {53/54, 0}, {55/56, 0}, {57/
  58, 0}, {59/60, 0}, {61/62, 0}, {63/64, 0}, {65/66, 0}, {67/68, 
  0}, {69/70, 0}, {71/72, 0}, {73/74, 0}, {75/76, 0}, {77/78, 0}, {79/
  80, 0}, {81/82, 0}, {83/84, 0}, {85/86, 0}, {87/88, 0}, {89/90, 
  0}, {91/92, 0}, {93/94, 0}, {95/96, 0}, {97/98, 0}, {99/100, 
  0}, {101/102, 0}, {103/104, 0}, {105/106, 0}, {107/108, 0}, {109/
  110, 0}, {111/112, 0}, {113/114, 0}, {115/116, 0}, {117/118, 
  0}, {119/120, 0}, {121/122, 0}, {123/124, 0}, {125/126, 0}, {127/
  128, 0}, {129/130, 0}, {131/132, 0}, {133/134, 0}, {135/136, 
  0}, {137/138, 0}, {139/140, 0}, {141/142, 0}, {143/144, 0}, {145/
  146, 0}, {147/148, 0}, {149/150, 0}, {151/152, 0}, {153/154, 
  0}, {155/156, 0}, {157/158, 0}, {159/160, 0}, {161/162, 0}, {163/
  164, 0}, {165/166, 0}, {167/168, 0}, {169/170, 0}, {171/172, 
  0}, {173/174, 0}, {175/176, 0}, {177/178, 0}, {179/180, 0}, {181/
  182, 0}, {183/184, 0}, {185/186, 0}, {187/188, 0}, {189/190, 
  0}, {191/192, 0}, {193/194, 0}, {195/196, 0}, {197/198, 0}, {199/
  200, 0}, {199/198, 0}, {197/196, 0}, {195/194, 0}, {193/192, 
  0}, {191/190, 0}, {189/188, 0}, {187/186, 0}, {185/184, 0}, {183/
  182, 0}, {181/180, 0}, {179/178, 0}, {177/176, 0}, {175/174, 
  0}, {173/172, 0}, {171/170, 0}, {169/168, 0}, {167/166, 0}, {165/
  164, 0}, {163/162, 0}, {161/160, 0}, {159/158, 0}, {157/156, 
  0}, {155/154, 0}, {153/152, 0}, {151/150, 0}, {149/148, 0}, {147/
  146, 0}, {145/144, 0}, {143/142, 0}, {141/140, 0}, {139/138, 
  0}, {137/136, 0}, {135/134, 0}, {133/132, 0}, {131/130, 0}, {129/
  128, 0}, {127/126, 0}, {125/124, 0}, {123/122, 0}, {121/120, 
  0}, {119/118, 0}, {117/116, 0}, {115/114, 0}, {113/112, 0}, {111/
  110, 0}, {109/108, 0}, {107/106, 0}, {105/104, 0}, {103/102, 
  0}, {101/100, 0}, {99/98, 0}, {97/96, 0}, {95/94, 0}, {93/92, 
  0}, {91/90, 0}, {89/88, 0}, {87/86, 0}, {85/84, 0}, {83/82, 0}, {81/
  80, 0}, {79/78, 0}, {77/76, 0}, {75/74, 0}, {73/72, 0}, {71/70, 
  0}, {69/68, 0}, {67/66, 0}, {65/64, 0}, {63/62, 0}, {61/60, 0}, {59/
  58, 0}, {57/56, 0}, {55/54, 0}, {53/52, 0}, {51/50, 0}, {49/48, 
  0}, {47/46, 0}, {45/44, 0}, {43/42, 0}, {41/40, 0}, {39/38, 0}, {37/
  36, 0}, {35/34, 0}, {33/32, 0}, {31/30, 0}, {29/28, 0}, {27/26, 
  0}, {25/24, 0}, {23/22, 0}, {21/20, 0}, {19/18, 0}, {17/16, 0}, {15/
  14, 0}, {13/12, 0}, {11/10, 0}, {9/8, 0}, {7/6, 0}, {5/4, 0}, {3/2, 
  0}} *)

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