What does recursive cosine sequence converge to?

Solution 1:

This is a standard trick worth knowing.

Supposing the limit does exist, call it $x$. If $x$ is the limit of the sequence, it has the property that $x = \mbox{cos}(x)$. From a graph, we can see that there is exactly one solution. Lastly, Wolfram Alpha tells us that $x = \mbox{cos}(x)$ has the solution $x = 0.739085$ as you said.

Solution 2:

This happens to be a relatively well-known number. It is called the Dottie Number, named after the not-at-all famous Professor of French, Dottie. I should also point out that the number is transcendental (also pointed out in the link).