Finding the complement of a function's domain

Question

I'm trying to find the list of x values not included in my domain of $0$ to $2\pi$. My current code is:

f[x_] := Cot[x]/Sin[x]
dom = FunctionDomain[{f[x], 0 <= x <= 2 π}, x, Reals]

NumberLinePlot[{f[x] > 0, f[x] < 0, f[x] == 0}, {x, 0, 2 π}, 
  Ticks -> {{0, π/2, π, 3 π/2, 2 π}}]

Thr above does give me the domain and a broken up plot, but it would be even better to just get the complement of the domain. Is this possible?

Answer 1

domcomplement = Reduce[Not @ dom, x, Reals]

x < 0 || x > 2 Pi || Element[x / Pi, Integers]

Show[Plot[ConditionalExpression[f[x], dom], {x, -π, 3 π}, 
  Frame -> True, Axes -> False, 
  FrameTicks -> {{Automatic, Automatic}, {{0, π/2, π, 3 π/2, 2 π}, Automatic}}], 
 NumberLinePlot[domcomplement, {x, -π, 3 π}, PlotStyle -> Red, Spacings -> {-15, 1, 1, 1}], 
 PlotRangePadding -> Scaled[.1]]