As the beam leaves the aircraft, the upper part of it is more horizontal than the lower part, which is probably around 45 degrees to the surface of the earth. The angle of depression when leaving the aircraft is therefore different for them (imagine looking at it from the side). Numbercrunching the angles with the square of the cosecant ensures that more of the signal is radiated in the top part of the beam to cover for the fact that when it reaches the surface it is attenuated over a wider area (because of the lower angle of depression). Thus the signal strength is more uniform.
The idea is not to waste energy when a normal fan beam radiates into space while you're looking on the ground - the cosecant beam has a more defined upper boundary, and the upside down version is used for ground mapping.
But this is old hat anyway - modern radars appear to have no difference between the shape of the beam in WX or MAP modes. We had an old Bendix on one of our 212s which mentioned it in the manual, but I have never seen it otherwise.
But then, many questions are way out of date, as witnessed by the symbols used for logic gates
