nLab cotangent function

Idea

In trigonometry the cotangent function is one of the basic trigonometric functions.

Definition

The contangent is the ratio of the cosine and the sine:

$$\cot(x) = \frac{\cos(x)}{\sin(x)}$$

Properties

Series expansion

The cotangent is the logarithmic derivative? of the sine function:

$$\cot x = (\log (\sin x))'.$$

Applying this observation to the Euler-Weierstrass product formula for the sine:

$$\sin (\pi x) = \pi x \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)$$

one obtains the following summation formula for the cotangent:

$$\pi\, \cot (\pi x) = \frac1{x} + \sum_{n=1}^\infty \left(\frac1{x + n} + \frac1{x - n}\right)$$

This expansion was used by Eisenstein as a starting point for developing the theory of trigonometric functions; Eisenstein’s account of elliptic functions, developed further by Weierstrass, Kronecker, and others, runs parallel to his trigonometric theory, as explained later by Weil. For some more details, see these notes by Varadarajan.

Related concepts

• tangent function

• trigonometric identity

References

• Wikipedia, Trigonometric functions – tan

• Veeravalli Varadarajan, Circular Functions (pdf).