He did say, however, (through Gandalf) that the Ring began to respond to Sauron's malice when it left him. Hence, it appears that Sauron was unable to "connect" with the Ring at anything but the most primal level. If he believed the Ring had been destroyed, he may also have believed that he had retained sufficient strength to reconstitute himself. Any initial efforts he might have made to determine if the Ring still existed apparently failed, since he didn't start searching for it until late in the third millennium. Hence, Sauron was simply too weak for most of the Third Age to know that the Ring existed, but he probably began to suspect something was not quite right as he continued to get stronger.

Gandalf knew that Sauron had taken a long time to reconstitute himself and to grow strong enough to declare himself openly again while the Ring existed but was separated from its master. It's not that much of a leap of faith to conclude that the destruction of the Ring should take Sauron so low he would be too weak to trouble Middle-earth again.

Sauron, according to Gandalf, didn't think anyone would consider destroying the Ring. Sauron's objective was to recapture the Ring before it could be used against him (which is what he expected). People are often blinded by their own expectations and reasoning to the reasoning of others.

That is what Gandalf counted on, and that is why Aragorn revealed himself to Sauron in the Palantir -- to feed that expectation. Whether Aragorn revealed himself as Thorongil is a guess. It makes more sense than revealing himself as a Ranger or a kid in Rivendell. Thorongil was a war-hero, a great captain. It would give Sauron reason to believe that he had now found the Ring and was returning to Gondor to begin using it.

At the very least we know that Aragorn did not reveal himself as he then appeared (at Helm's Deep).
------------------
Xenite.Org: Science Fiction and Fantasy