Answer
The sequence is convergent.
Work Step by Step
By the definition of the sequence,
$x_{n+1}$ is the greater value $(x_{n}$ and "something else").
So, $x_{n+1}$ is never smaller than $x_{n}$.
The sequence is monotonic (nondecreasing).
Since every term is $\cos$("something"), the sequence is bounded, below by -1, and above by 1.
By Th. 6, the sequence is convergent.
