Can I add recessed lights to a ceiling fan, without adding a new switch?
It is theoretically possible, but there are several problems:


The pull chain switch for the lights has a certain amperage rating and adding the cans might exceed that rating
The fan assembly with switch is likely a UL approved device, and modifying it would likely void that approval, which may run afoul of local code
You would need to run the hot wire for the cans into the fan unit to join it to the hot coming out of the pull chain switch. Or you would need to run a pigtail from that switch to the canopy to meet the wire for the cans. Either case requires disassembly of the fan unit and switch unit.
You may have a space problem fitting another wire and wire connector in the area of the pull chain switch.


Probably could be done, but why not rely on a separate switch on the wall? Or a smart switch with a remote?