From Förberedande kurs i matematik 1

If we use the unit circle and mark on the angle \displaystyle \pi , we see immediately that \displaystyle \text{cos }\pi \text{ }=-\text{1 } and \displaystyle \text{sin }\pi \text{ }=0.

Thus,

\displaystyle \tan \pi =\frac{\sin \pi }{\cos \pi }=\frac{0}{-1}=0