The two angles I assigned were done that way to get the data to follow the equation. I could have worked out a different way, but I wasn't that sure the cone angle for the bare LED was determined the same way as with the optics. And, by then I was getting tired of trying to find another rational way to reconcile the data. Every other method I tried failed. The angle I'm looking for is the half angle of the cone, 30 degrees for a 60 degree cone, for example. Then, since most of the light is in the central half of that angle, I'm using 1/4 of the cone angle in the equation. That, when squared is proportional to the area of most intense light from the LED. The correlation doesn't work nearly as well using 1/2 or one times the cone angle. (The tangent goes negative with angles larger than 90 degrees.)
If you look at the Cree data in their pdf, the shape of the light intensity vs angle is considerably different for the XM-L and XR-E. And, I have no idea what the shape with the optics looks like, but I understand that it is best to work with only the inner half of the angle.
I may try to see what it takes to fit the data I have from Ebay cheap 3 watt LEDs to that equation, just to get a better feel for whether or not the equation will work with other LEDs.
One problem with trying to derive an equation from a bunch of data is the errors in the data. The PAR meter isn't accurate below 10, since it doesn't read in decimals, only whole numbers. And, my current measurements are probably riddled with errors since it was very hard to keep the multimeter probes in the right locations - probably a +/- 10% error at best. Then, when you square things the errors double. And, human errors in writing down data enter into it too.