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Find the Fixed Points of ArcCoth
ArcCoth has two real fixed points.
In[1]:=

Click for copyable input
fps = Reduce[ArcCoth[x] == x, x, Reals]
Out[1]=



In[2]:=

Click for copyable input
Plot[{ArcCoth[x], x}, {x, -3, 3}, 

 Epilog -> {Red, PointSize[0.025], Point[{x, x}] /. {ToRules[fps]}}]
Out[2]=