The graphs of the tangent and cotangent functions show some interesting patterns. Let’s break these down:

1. Periodicity:

   • Both the tangent and cotangent functions repeat every $\pi$ radians.
   • This means that if you look at the values of these functions, they will be the same after every $\pi$ radians.

2. Asymptotes:

   • The tangent function has lines (called vertical asymptotes) that it never touches at $x = \frac{\pi}{2} + k\pi$. Here, $k$ is any whole number.
   • For the cotangent function, the vertical asymptotes are at $x = k\pi$ for whole numbers $k$.

3. Range:

   • Both tangent and cotangent can take any value from negative to positive infinity. In other words, there are no limits on the output values.

4. Symmetry:

   • The tangent function is called an odd function. This means that if you plug in a negative number, you get the opposite output: $f(-x) = -f(x)$.
   • The cotangent function is also an odd function, following the same rule: $f(-x) = -f(x)$.

These features help us understand how tangent and cotangent behave on a graph.