Is $\cos(x[n])$ stable?
The I/O relation
$$y[n] = \cos(x[n])$$

represents a stable system. 

Because by definition of $\cos(x)$, irrespective of the input $x[n]$, its output will always be bounded by $$-1 \leq \cos(x[n]) \leq 1 $$ for all $x[n]$ for all $n$. Even if $x[n]$ is undefined ( such as would be the case with $x[n] = 1/n$ at $n=0$ as Marcus pointed out) we for sure know that the system output is bounded (the value of input is undefined, but the output is known to be within $\pm1$.)