Figure 1.5.7(a) contains a graph of $\tan \left(x\right)$ and its vertical asymptotes that appear to be at $x\=\frac{2 k\+1}{2}\mathrm{\π}$, where $k$ is an integer. Writing $\tan \left(x\right)$ as $\frac{\sin \left(x\right)}{\cos \left(x\right)}$ shows that the vertical asymptotes occur at the zeros of $\cos \left(x\right)$, namely, at the odd half-multiples of $\mathrm{\π}$, that is, at:

$\dots \,-\frac{5 \mathrm{\π}}{2}\,-\frac{3 \mathrm{\π}}{2}\,-\frac{\mathrm{\π}}{2}\,\frac{\mathrm{\π}}{2}\,\frac{3 \mathrm{\π}}{2}\,\frac{5 \mathrm{\π}}{2}\,\dots$