This is a weird definition of ArcTan, certainly not a standard one. Normally, arctan is a function of one variable, not two, and arctan(x) basically means the angle whose tangent is x. I'll refer to this function of one variable as arctan (with lower case a and t) to distinguish it from that function ArcTan of two variables.

If you work out sin(12.0321°) * cos(23.45°), it comes to 0.1912425; and cos(12.0321°) is 0.9780309. So we are told that ArcTan(0.1912425,0.9780309) = 11.0639°. What I found by trial and error is that if I divide 0.1912425 by 0.9780309 it comes to 0.1955383. Then when I press the button on my calculator (remember that is just another name for arctan), it gives the answer as 11.0639°, which is what we want.

So it seems to me as though your ArcTan is related to the conventional arctan by the relation ArcTan(x,y) = arctan(x/y).