Answer
secant function
Work Step by Step
Among the six trigonometric functions, sine, cosine, and their respective reciprocal functions, cosecant and cotangent, have $2\pi$ as periods.
Note that the secant function is undefined for $x=..., -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, ...$
Note that $x=(2n+1)\frac{\pi}{2}$ will always involve an odd number coefficient for $\pi$ for any integral value of $n$
Therefore, the function that satisfies the given descriptions is the secant function.
