The Cotangent Function

Definition For a real number $x$, measured in radians, with $\sin x \ne 0$, the cotangent of $x$ is

On the unit circle the angle $x$ gives the point $(\cos x, \sin x)$, so cotangent is the horizontal coordinate divided by the vertical coordinate of that point. The only condition is that the denominator $\sin x$ is not zero. Nothing about $\cos x$ needs to be checked.

Definition Cotangent is undefined exactly when $\sin x = 0$, which happens exactly at the integer multiples of $\pi$. Its domain is

This domain comes straight from the denominator of $\frac{\cos x}{\sin x}$. It is not the same as the domain of tangent, which excludes $\frac{\pi}{2} + k\pi$ instead.

Definition A zero of cotangent is a number $x$ in its domain with $\cot x = 0$. A fraction is zero exactly when its numerator is zero and its denominator is not, so $\cot x = 0$ exactly when $\cos x = 0$ and $\sin x \ne 0$. Since $\cos x = 0$ forces $\sin x = \pm 1$, the zeros are

Definition For every $x$ in the domain, $x + \pi$ is also in the domain and

because adding $\pi$ moves the unit circle point to the opposite point. So $\pi$ is a period. It is the least positive period. Suppose some $p$ with $0 < p < \pi$ were a period. Since $\frac{\pi}{2}$ is in the domain, so is $\frac{\pi}{2} + p$, and $\cot\left(\frac{\pi}{2} + p\right) = \cot \frac{\pi}{2} = 0$. That would make $\frac{\pi}{2} + p$ a zero of cotangent strictly between $\frac{\pi}{2}$ and $\frac{3\pi}{2}$, but the zeros are spaced exactly $\pi$ apart and none lies there. Cotangent is also odd. For $x$ in the domain, $-x$ is in the domain too, and since cosine is even and sine is odd,

Definition The line $x = a$ is a vertical asymptote of the graph of $f$ if $f(x)$ grows without bound, toward $\infty$ or $-\infty$, as $x$ approaches $a$ from at least one side. The graph of cotangent has a vertical asymptote at every excluded input $x = k\pi$. Near $x = k\pi$ the numerator $\cos x$ is close to $\pm 1$ while the denominator $\sin x$ is close to $0$, so the quotient is huge in size. Just to the right of each $k\pi$ the values go to $\infty$, and just to the left they go to $-\infty$.

Definition The graph of $y = \cot x$ consists of separate branches, one on each open interval $(k\pi, (k+1)\pi)$ with $k$ an integer. On each branch cotangent is continuous and decreasing. On $(0, \pi)$ this follows from what sine and cosine do there.

• On $\left(0, \frac{\pi}{2}\right]$ the numerator $\cos x$ decreases and stays $\ge 0$, while the denominator $\sin x$ increases and stays positive. A smaller top over a larger bottom gives a smaller quotient, so $\cot x$ decreases.
• On $\left[\frac{\pi}{2}, \pi\right)$ write $\cot x = -\frac{\lvert \cos x \rvert}{\sin x}$. Here $\lvert \cos x \rvert$ increases from $0$ while $\sin x$ decreases and stays positive, so the fraction grows and its negative decreases.

The two pieces share the input $\frac{\pi}{2}$, so cotangent decreases on all of $(0, \pi)$. Every other branch repeats this one, because $\cot(x + k\pi) = \cot x$. Each branch falls from $\infty$ near its left endpoint to $-\infty$ near its right endpoint, crosses the axis once at its midpoint $\frac{\pi}{2} + k\pi$, and takes every real value exactly once. The range of cotangent is all of $\mathbb{R}$.

Definition For every real $x$ with $\sin x \ne 0$ and $\cos x \ne 0$, that is, $x \ne \frac{k\pi}{2}$ for every integer $k$,

Both sides are defined together only on this set. At $x = \frac{\pi}{2} + k\pi$ the left side is $0$ while the right side is undefined, because $\tan x$ itself does not exist there. So the identity is a statement about the common domain, not a definition of cotangent.