Cotangent
Cotangent is the reciprocal of tangent, defined as the ratio of the adjacent side to the opposite side in a right triangle.
Formula
\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}} = \frac{\cos(\theta)}{\sin(\theta)}
Definition
Cotangent (written "cot") is the flip of tangent: $\cot(\theta) = \cos(\theta)/\sin(\theta) = \text{adjacent}/\text{opposite} = 1/\tan(\theta)$. It is undefined when $\sin(\theta) = 0$, and the Pythagorean identity involving cotangent, $1 + \cot^2(\theta) = \csc^2(\theta)$, comes from dividing $\sin^2 + \cos^2 = 1$ by $\sin^2$. Cotangent has period $\pi$, the same as tangent, with poles at every integer multiple of $\pi$ and Laurent series $\cot(x) = 1/x - x/3 - x^3/45 - 2x^5/945 - \ldots$
Example
If $\tan(45^\circ) = 1$, then $\cot(45^\circ) = 1/1 = 1$; and $\cot(60^\circ) = \cos(60^\circ)/\sin(60^\circ) = (1/2)/(\sqrt{3}/2) = 1/\sqrt{3} = \sqrt{3}/3 \approx 0.577$. Its derivative is $d/dx[\cot(x)] = -\csc^2(x)$, and the integral of $\cot(x) = \ln|\sin(x)| + C$.
Key Insight
Cotangent is like tangent looking at the angle from the other side of the triangle. Euler's partial fraction expansion $\cot(\pi x) = 1/x + \sum (1/(x-n) + 1/(x+n))$ over positive integers $n$, combined with evaluating at $x = 1/2$, provides an elegant proof that $\sum 1/n^2 = \pi^2/6$, solving the Basel problem.