f[[Theta]_] =
 Integrate[1/(Cot[[Theta]]^2 - Sin[[Theta]]^2)^((1/2)), [Theta]]

-(1/Sqrt[Cot[[Theta]]^2 - Sin[[Theta]]^2])
 2 (-9 + 4 Sqrt[5]) Cos[[Theta]/2]^4 Csc[[Theta]] EllipticF[
   ArcSin[Sqrt[9 + 4 Sqrt[5]] Tan[[Theta]/2]^2],
   1/(9 + 4 Sqrt[5])^2] Sqrt[9 + 4 Sqrt[5] - Tan[[Theta]/2]^4] Sqrt[
  1 - (9 + 4 Sqrt[5]) Tan[[Theta]/2]^4]

InverseFunction[f][1]

[Theta]1[t_] := NSolve[f[[Theta]] == t, {[Theta]}]
[Theta]1[1]