The trick is to use a farm jack (the same as a jeep Hi-Lift jack).
They are available at Harbor Freight. I haven't seen them at Home Depot or Lowes.
Most attach a chain to the jack, but I used a 7' bike security cable with both ends wrapped around the stump.
If it is not too big you can do it with the jack alone, but it tends to pull to one side, so a tripod with a couple of 2x4's bolted to the top is better.
The jack was rated at 2 1/2 tons for winching and test to 3 1/2 tons and I had to use most of my weight to break the 2 inch taproot a foot down.
Another option would have been to dig a wider hole down a foot so you could get under the stump and saw the tap root off.
A taproot is a large, central, and dominant root from which other roots sprout laterally. Typically a taproot is somewhat straight and very thick, is tapering in shape, and grows directly downward. A carrot is the taproot of a carrot plant. Dandelions have one thick taproot that can extend more than a foot into the ground.
In most plants it is replaced later in the plant's development by a fibrous root system.
Most trees start with a tap root but the main roots spred out horizontally once they are established. Trees such as walnuts, pecans, oaks, elms, pines and firs, maintain a taproot, but most fruit trees, almonds and many shade trees grow a fibrous root system. This is composed of a number of horizontal roots radiating out from the trunk.
Soil characteristics strongly influence the architecture of taproots; for example, deep rich soils favour the development of vertical taproots.
One of the videos below got the roots of a 30' cyprus out by getting the jack under a couple of the large horizontal roots.