There is an old saying that goes like this: When man is gone from the face of this earth, there will still be coyotes and cockroaches...
Dogs have been protecting sheep (herds) for hundreds of years and do a pretty awesome job. That is one answer to the probelm. Government intervention, traps and/or poison, is INDESCRIMINANT. Experience has taught me that I'd only use this as a very last resort. Plus, no government 'program' has ever eliminated a coyote or couger problem. They only eliminate a few and scare the rest away until a more opportunistic time. The coyotes WILL be back...
My qualification in these thoughts is, I trapped for a (winter) living for many years and was associated with both government trappers and gov programs concerning sheep ranches. Also, my neighbors currently (try to) raise sheep, and coyotes have pretty much ruined their operation. Yes, coyotes attack live sheep and have even attacked the neighbor's Austrailian Shepard that 'protects' them, hence BIG dogs for that end of the operation.
I use to think along the lines of "night-vission" devices to combat the problem, but no one can be that vigilant, as coyotes attack at both day and night.
I, Myself, and Oregon DF+W, agree that electric fencing is the best deterant. If the coyotes are getting through what you already have in place, INCREASE the amount of wire by decreasing the spaces inbetween. I use 4' of 'no-climb' kennel wire on the bottom topped with another 4 feet of electified wire (6" spacing) for a total of 8' feet in heigth. Any less and you are asking for failure.
We also have coons and skunks, and that requires another 2 or 3 strands of electified wire at the bottom, inside and outside of the perimeter, with deligent routine weed-eating, as a single strand of grass can ground out, or even start a fire in the dry season.
This approach is relatively expensive, but not as expensive as both the loss of livestock and the broken hearts you suffer over the years when you have had to put down a prize animal that is laying there half-butchered by a coyote or couger attack.
This system will work against coyotes, couger, deer and the smaller vermin that would like to at least come in for free ducks, chickens, eggs and whatever else is on their minds. No one has ever lost an animal that has a good, well maintained perimeter wire system in place, but it still takes vigilance and up-keep or the preditors will find a way around or under it, some way, some how...