Ignoring the derivative which does show $f(x) = \cos(1/x)$ is increasing when $\sin (1/x)/x^2$ is positive (which is the case for all $x > 1/\pi$ [which is the case for all natural numbers n]), "common sense" should indicate $f(x) = \cos(1/x)$ is increasing for significantly large $x$.
As $x \rightarrow \infty$ then $1/x$ decreases to 0 (as a limit, of course). As $v = 1/x$ decreases from $v = \pi$ ($x = 1/\pi$) to 0, $\cos v$ increases from -1 to 1 (as a limit, of course).
So $f(x)$ is monotonically increasing on $[1/\pi, \infty)$.
But $f(x)= \cos(1/x)$ is monotonically decreasing on $(-\infty, - 1/\pi]$. (But then it oscilates innumerably, is undefined at x = 0, oscilates innumerable times between 0 and $1/\pi$ and then increases in value from -1 to a limit of 1.)