Answer
cosecant
Work Step by Step
Among the six trigonometric functions, sine and cosine and their respective reciprocal functions, cosecant and secant, have a period of $2\pi$.
Having $x\ne n\pi$ where $n$ is an integer means that the function is not defined for multiples of $\pi$ including $0$.
Note that the cosecant function has, as its vertical asymptotes., the lines $x=...-3\pi, -2\pi, -\pi, 0, \pi, 2\pi, 3\pi, ...$.
Therefore, the function that satisfies all the dscription is the cpsecant function.
