I'm exploring the properties and behaviors of trigonometric functions, specifically the cotangent function $\cot(\pi x) $ and its behavior near its discontinuities. The normal domain of $\cot(x)$ excludes integer multiples of $\pi$, where the function is traditionally undefined due to the sine function being zero:
$$ x \in \mathbb{R} \setminus \{k\pi \mid k \in \mathbb{Z}\} $$
At these points, $x = k\pi$, the function approaches negative or positive infinity depending on the direction of the approach:
$$ \lim_{{x \to k\pi^+}} \cot(\pi x) = -\infty \quad \text{and} \quad \lim_{{x \to k\pi^-}} \cot(\pi x) = +\infty $$
From this, we can see that as $x$ approaches $+\infty$, the angle $\theta$ whose cotangent is $x$ must approach $0$:
$$ \lim_{{x \to +\infty}} \operatorname{arccot}(x) = 0 $$
Considering this, I'm curious if this expression is accurate in context:
$$\operatorname{arccot}(\cot(\pi k))=0$$
Normally, we would consider $\cot(\pi k)$ undefined for integer $k$. However, if we were to extend the definition to infinity at these points, it seems we could argue that $\operatorname{arccot}(\cot(\pi k)) = 0$.