### Instruction: Find the equation of the asymptote of the graph of $r = \cos 2 \theta \sec \theta.$.### Response: I know that an asymptote of a polar graph occurs when the polar function $r$ is undefined, which means that the denominator of any fractional expression must be zero.
In this case, the only fractional expression is $\sec \theta = \frac{1}{\cos \theta}$, so the asymptote occurs when $\cos \theta = 0$.
I also know that the equation of an asymptote of a polar graph is $\theta = \alpha$, where $\alpha$ is a constant angle.
Therefore, I need to find the values of $\theta$ that make $\cos \theta = 0$ and write them as constants.
I recall that $\cos \theta = 0$ when $\theta$ is an odd multiple of $\frac{\pi}{2}$, so $\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots$ or $\theta = -\frac{\pi}{2}, -\frac{3\pi}{2}, -\frac{5\pi}{2}, \dots$
However, since $\theta$ is periodic with period $2\pi$, I can simplify these values by adding or subtracting multiples of $2\pi$ and obtain the general form $\theta = \frac{\pi}{2} + n\pi$, where $n$ is any integer.
Therefore, the equation of the asymptote is $\theta = \frac{\pi}{2} + n\pi$.