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