Welcome to the forums and congrats on the purchase of your new Cougar.
There are different ways to flush a black tank. You may be able to do all of these and maybe only one or two of them. I'll give you different methods.
1) The simplest is to simply drain your black tank. Open the black tank valve under the camper and let it drain completely. If you install a clear section of pipe (you can get these at Wall Mart), you can see what's actually coming out.
When it quits draining, you are basically done. But, if you want to go further, then close the outside valve, go back inside your camper and simply push the toilet peddle and fill the black tank again. When it's full, dump it from the outside again.
This can be done on any camper or RV.
2) Black tank flusher built into your camper. If you have a black tank flush hook-up, then you .... a) drain the tank as above in #1. Then attach a garden hose to the black tank flusher, turn on the garden hose, and water will spray inside your black tank pushing the remaining contents out. You can run the water as long as you need, until it runs clear (using that same clear attachment on our outside valve).
3) Use a black tank flusher wand:
Click here. You attach this to a garden hose and simply insert it into the toilet bowl. Hold the foot pedal down, turn the water on and let it spray the inside of the black tank. Again, once the water runs clear from the outside outlet (again that clear section), you know it's rinsed out pretty good.
4) Another method is to back flush. In order to back flush, both the grey and the black water tanks need to exit at the same outlet. Before doing this, you need to attach a 3rd valve, like a Flush King
Click here. The steps may sound awkward, but it's really very simple.
a) attach the flush king.
b) drain the black tank.
c) close the flush king valve, leave the black valve open and open the grey water. This will allow the grey water (shower and bathroom sink) to back flush into the black tank.
d) count to 20, then close the grey tank valve so no black water will flow into the grey tank.
e) open the flush king again and drain the black tank. Again, that clear section, you'll see what's coming out.
f) depending on how much water you have in your grey tank, I can usually back flush 3 times in a row doing this. That leave enough gray water in the gray tank for a final rinse of the sewer hose to flush out any remaining black water.
5) one more, the flush king. On the flush king, you can also attach a garden hose and fill the black (or the grey) tank. This is also a "back flush" method that works very well.
There is really nothing mysterious about flushing the black tank. In short, all you are doing is adding cleaner water and dumping it again, which will pull out more "stuff" that sticks behind.
Again, welcome to the forums, congrats again! And Happy Camping!