Answer
Even, symmetric to the y-axis.
Work Step by Step
We know that $\sin$, $\csc$, $\tan$ are odd functions, which means $f(-\theta)=-f(\theta).$
We know that $\cos$, $\sec$, are even functions, which means $f(-\theta)=f(\theta).$
We know that if a graph is symmetric to the y-axis, then the points $(a,b)$ and $(-a,b)$ will be on the graph, hence $f(-b)=f(b)$, hence this is the same as the function being even.
We know that if a graph is symmetric to the origin, then the points $(a,b)$ and $(-a,-b)$ will be on the graph, hence $f(-b)=-f(b)$, hence this is the same as the function being odd.
We know that if a graph is symmetric to the x-axis, then the points $(a,b)$ and $(a,-b)$ will be on the graph, but then for $a$ there are two different values, hence this cannot happen for a function.
We know that for the $\cos$ function, $f(-\theta)=f(\theta)$, hence it is even. Therefore its graph is symmetric to the y-axis.
