Let $F(x,y)=\cos y\hat{i} + \cos x\hat{j}$. Then its vector lines satisy $-\sin x+C =\sin y$ or $\sin x + \sin y = C$. First, I try to guess how it looks like by considering $\cos(t)$ is positive, decreasing when $t \in [0,\pi/2]$ and it is negative, decreasing when $t \in [0,\pi/2]$ and so on. However, it is hard to get an appropriate graph. Then I try to guess how vector lines look like but $\sin x + \sin y = C$ is also hard to imagine.
So, how can I sketch the graph of $F$ and its vector lines?
This is supposed to be a comment but for some strange reason, I cannot input this correctly through the comment box.
I don't know what are vectors lines. It seems your vector lines ($\sin x + \sin y = C $ ) are the orthogonal trajectories for the family of streamlines / fieldlines associated with vector field $F = (\cos(y),\cos(x))$.
On WA, you can plot the streamlines of $F$ using the command
StreamPlot[{Cos[y],Cos[x]},{x,-3,3},{y,-3,3}]. To obtain the orthogonal trajectories, you need to manually rotate the vector field by $90^\circ$. i.e use the commandStreamPlot[{-Cos[x],Cos[y]},{x,-3,3},{y,-3,3}]instead.If you have Mathematica, there are more powerful variants of this sort of plot. Other CAS should have similar functionalities but I don't know the details.