f = 1/Tan[1/t];
Plot[f, {t, - Pi/2, Pi/2},
 ExclusionsStyle -> {None, Directive[Red, AbsolutePointSize[5]]}]

singularityDomain[f_, x_] := Module[{res = FunctionDomain[f, x]},
  Reduce[! res] /; ! MatchQ[res, _FunctionDomain]]

Clear[x]
singularityDomain[1/Tan[1/x], x]