Absolutely! They can both be combined.
So Rimadyl (you got it right) is an NSAID... sort of like aspirin, more like ibuprofen, but SAFER for dogs. Dogs can't take ibuprofen, it's toxic
to them. And aspirin is likely to cause stomach
ulcers when given for a long time.
Rimadyl targets very specific pain receptors in a dog, and therefore treats
the pain better than other meds can. For that reason, I think it is pretty cool.
The natural medications are anti-inflammatory too, but nobody is really sure how. They also help to increase flexibility of ligaments, increase joint fluid for cushion, and strengthen the cartilliage found inside the joints.
In this way, fish oil can be supplemented as well. It seems to help too. Doses are hard to determine, but most commercial fish oil will have doses listed right on the bottle.
After having your pet on these natural meds for 2-3 months you could try weaning
off of the Rimadyl to see if it's needed. Some pets with early arthritic changes will do just fine on the Cosequin without the Rimadyl. Go slowly though when reducing the Rimadyl, because you might find that you can get it to work at a very low level (maybe 1/2 or 1/4 of what you expect) when given with the other meds..
I always tell people to try to wean to the lowest effective dose to save on cost of meds, and to protect the pets liver over time. On harder days you can always go back to the prescribed dose.