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I am using the following Solve code:

Assuming[cosαcrit > 0 && cosαcrit <= 1, 
 Simplify[y /. 
   Solve[(cosαcrit - Cos[x]*Cos[y])/(Sin[x]*Sin[y]) == -1, 
     y][[1]]]]

The output is something like

ConditionalExpression[ArcTan[…,…]+2*Pi*c1,c1 $\in$ Z]

I have two questions:

1) How to force c1 to be zero in order to remove ConditionalExpression brackets

2) What is the interpretation of the comma inside ArcTan, and how to convert the output to traditional form?

I would be very appreciated if you will help me.

P.S. The first question is already not relevant.

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You can get rid of the C[1] with a replacement rule, and this also eliminates the ConditionalExpression. You can convert ArcTan[x, y] to the mostly equivalent form ArcTan[y/x] also with a replacement rule:

Assuming[
    cosαcrit>0&&cosαcrit<=1,
    Simplify[y/.Solve[(cosαcrit-Cos[x]*Cos[y])/(Sin[x]*Sin[y])==-1,y][[1]]]
] /. C[1]->0 /. ArcTan[x_,y_]:>ArcTan[y/x]

-ArcTan[((cosαcrit + Cos[x] Sqrt[-(-1 + cosαcrit^2) Csc[x]^2]) Sin[x])/( cosαcrit Cos[x] - 1/Sqrt[Csc[x]^2/(1 - cosαcrit^2)])]

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