Cougars are not hard to kill, I've taken them at close range with .22 mag, and seen them taken at close range with .22 mag, .32 H&R mag and 357 Mag. With a 44 caliber, shoot either special or mag, the load should not matter at tree'd cat distances so long as you make a good shot. Make a bad shot, and it don't really matter what you use, just get ready for some tore up dogs. I personally would use a good hard cast with a 44, just so the holes that the taxidermist has to sew up ain't so big.
