These are all complex solutions. There might be more if we take `ArcSin` and `ArcCos` as multi-functions.

    sol = {9/5 - (2 I Sqrt[11])/5, 
       1/5 - (2 I)/5, -1, -3 - (2 I)/Sqrt[5], -(23/5) - (2 I Sqrt[19])/5, 
       9/5 + (2 I Sqrt[11])/5, 
       1/5 + (2 I)/5, -(9/5), -3 + (2 I)/Sqrt[5], -(23/5) + (
        2 I Sqrt[19])/5};

    With[{x = a + I b}, 
     ContourPlot[{Re[2 ArcCos[Sin[Pi*x]] - Pi*Cos[ArcSin[x + 1]]] == 0, 
       Im[2 ArcCos[Sin[Pi*x]] - Pi*Cos[ArcSin[x + 1]]] == 0}, {a, -10, 
       10}, {b, -4, 4}, PlotPoints -> 100, Epilog -> Point@(ReIm /@ sol)]]

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[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/8xih0.png