Here is a simplified version of your result which I think explains the two kinds of commas which appear.
In[1]:= Solve[EG+(CD+DE)Cos[\[Delta]]+BC Cos[\[Alpha]]+BG Cos[\[Gamma]]==0 &&
(CD+DE)Sin[\[Delta]]+BC Sin[\[Alpha]]+BG Sin[\[Gamma]]==0, {\[Alpha],\[Gamma]}]
Out[1]= {{\[Alpha]->ConditionalExpression[ArcTan[numerator1,denominator1]+2 \[Pi] C[1],C[1] \[Element] Integers],
\[Gamma]->ConditionalExpression[ArcTan[numerator2,denominator2]+2 \[Pi] C[2],C[2] \[Element] Integers]},
{\[Alpha]->ConditionalExpression[ArcTan[numerator3,denominator3]+2 \[Pi] C[1],C[1] \[Element] Integers],
\[Gamma]->ConditionalExpression[ArcTan[numerator4,denominator4]+2 \[Pi] C[2],C[2] \[Element] Integers]}}
There are two different forms of ArcTan in Mathematica, ArcTan and ArcTan[x,y] where the second form can express exactly which quadrant the point is in.
That accounts for the four commas inside the four ArcTan[] expressions.
ConditionalExpression[Value, BooleanConditionWhichMustBeTrueForThisToBeTheValue] accounts for the four commas outside the ArcTan.
I think that accounts for all of the commas.