What methods can a person use to safely repair a roof that is slanted at a 45 degree (or greater) angle without using Roof Jacks? Is there anyway it is possible for a person to safely walk on such a roof or would the chance of a fall be too great?
You wear a harness attached to a rope that is thrown over the roof and tied to something solid such as a tree.
The harness kit may look like this:
The harness is most often used in a manner as shown here. A short connecting rope goes from the middle of the high back over to a D-Clip that then attaches to the main safety rope via a Robe Grab.
Rope grabs come in various styles (one shown below) but are basically designed to permit them to slide along the rope when they are directly pushed but will clamp tightly on the rope when pull tension is applied to the connecting ring from the D-Clip and short rope to the roofer.
This picture shows a harness in use on a roof....albeit not quite as steep as the OP has asked about.
Here is another in use application.
When young and immortal, you [may think you] can climb around on a 45 degree roof with a pair of sneakers and the confidence that you'll never fall off. Mortality may intervene if you do fall off, and as you get older and mortality has more of a grip precautions are [more obviously] advisable...
Roof Jacks are commonly available for rent (by day, week or month) cheaply - they also can be purchased. Depending on the nature of the work needed, pump jack (or plain) staging set up along the wall and raised near the roof edge may provide better working access to the lower edge of the roof and a thing to catch you if you slide down the roof.
A ladder hook (they can also often be rented) can also be used to hang a ladder from the peak, if the scale of repair is small. The ones with wheels allow you to roll up on the wheel and then flip it over to hook the peak. Old style was just slide the ladder up and flip it, perhaps using a burlap sack to prevent the end of the ladder from dinging the roof.
With a rope over the roof, you need to be careful that it cannot slide off the end, and you need to be sure about your knots attaching the rope to a fixed object (try a climbing site and the "figure 8 follow through") - your life depends on this, so get it right. Also be careful of anything (like sharp edges on metal roofing or flashing) that could cut the rope.