That zero test can be done by hand, by showing that the two arcs have the same tangents (and realizing they lie in the same quadrant).
ResourceFunction["RadicalDenest"][
Together[TrigExpand[Tan[2 ArcTan[(2 - Sqrt[3])^(3/2)]]]]]
During evaluation of In[639]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
(* Out[639]= Sqrt[2]/5 *)
Tan[ArcCot[5/Sqrt[2]]]
(* Out[640]= Sqrt[2]/5 *)
So that resource function is a bit noisy. No doubt one of us is to blame.
