I too have a 45 degree slope that I keep mowed as a fire break. We live in a forest. I modified a rear wheel drive walk behind mower by adding a steel circle around the drive tires. I put 3/4 inch screws through the steel hoop that act like large studs. Doing this puts much more stress on the drivetrain of the mower since there is virtually no more slippage of the wheels. If the mower has nylon gears cut into the drive wheels, it will fail within 4 hours of use. Trust me, I know from experience! 
You need a mower that has a chain drive gear reduction transmission. This type mower lasted 1 year before the chain/gear transmission failed. I replaced the transmission assembly and added a grease fitting to the housing around the chain drive. Each spring I pump it full of grease and it has worked for 4 years now with no failures. You will need replacement clutch gears and springs as these fail once or twice a year. These are located behind the drive wheels.
I mow on a diagonal so the mower is less likely to flip over. It takes some strength. I have built a robotic mower for this hill that is R/C controlled. My problem with it is traction. When there is no person attached to steer and support the mower, steering is difficult. On more level ground, my robot works very well. It is a hybrid design. There is a gas engine for the mower and a 24 volt alternator. There are two 12 volt batteries powering 2 electric motors. There is an electronic speed controller which is controlled by the radio system. It also has steel wheels with large studs for traction. I am looking into a tracked drive train but so far, they are very pricey!
I just thought I would add my experience to this topic.
You need a mower that has a chain drive gear reduction transmission. This type mower lasted 1 year before the chain/gear transmission failed. I replaced the transmission assembly and added a grease fitting to the housing around the chain drive. Each spring I pump it full of grease and it has worked for 4 years now with no failures. You will need replacement clutch gears and springs as these fail once or twice a year. These are located behind the drive wheels.I mow on a diagonal so the mower is less likely to flip over. It takes some strength. I have built a robotic mower for this hill that is R/C controlled. My problem with it is traction. When there is no person attached to steer and support the mower, steering is difficult. On more level ground, my robot works very well. It is a hybrid design. There is a gas engine for the mower and a 24 volt alternator. There are two 12 volt batteries powering 2 electric motors. There is an electronic speed controller which is controlled by the radio system. It also has steel wheels with large studs for traction. I am looking into a tracked drive train but so far, they are very pricey!
I just thought I would add my experience to this topic.
