I use this strategy, and it works:
First of all, you have already noted that almost every big cat in ZT gets angry with almost every thing you do to its environment, right?
1.For example, for the Bengal Tiger, i build a big exhibit and fill it with the Rainforest Floor terrain, then i put the Tiger and start making changes to the floor. (It works for almost every big cat since it gives him a happiness boost from the beginning due to the floor).
2. If you have Snow Leopards or Pumas, first, do the same thing to the floor, then proceed to build some hills and cliffs before placing the animal in the exhibit.
3. After that, put the shelter (always the rock cave, or if you don´t research it yet, use the concrete shelter) and you should be on the right way.
4. Then, put the rocks and trees. But not the same kind of tree or rock; for example, i like to do rainforest-like exhibits to tigers, so i put the Rainforest Rock Formation, the Rock with the Waterfall and then use lots of medium common rocks.
For the trees is the same thing, with Tigers, i use mangrove, palm tree, sometimes durian tree and the rainforest fern. But try to place 4 of them in a square; and a lot of these. It usually boost the happiness of the suitability for the animal to 93-100.
5. Then put other animals for company. If another one´s needed.
6.Finally, use the toys. But, for example, in case of Black Leopards and almost every leopard, use one if you have two animals.
Bonus: It´s hard to keep a continous happiness level with tricky cats like the Puma, Snow Leopard, Jaguar and Black Leopard.
Here´s some tips for them:
-Sunk the exhibit two tiles.
-Always, use rock fences with the puma and the Snow Leopard. But only with one visible side (big cats hate people look at them from almost every side).
-The same for the jaguar, but with this guy, you should use a concrete fence with the plexiglas or the half concrete/half chain link fence.
Well, if you follow these tips in the right order, you can boost the animal happiness without pausing or selling and buying again (well, except for the Puma and the Snow Leopard, they are hard).
I hope this mini-guide helps someone.