duplicate the original circle layer, and move the copy below the latitude lines layer
unlock this layer
select the circle and the lines and hit the Outline filter on the Pathfinder panel
the lines will seem to disappear - they haven't, the stroke has just been set to 0 so...
set the stroke and fill back to default white fill and black stroke, then clear the fill again
(or just set the stroke to 1px black)
this has effectively broken the lines where they intersect the circle
use the path selection tool (White arrow) select the line sections which extend above the circle)
hit delete twice to remove them
do the same for the line sections below the circle - you should have:
