In trigonometry the cotangent function is one of the basic trigonometric functions.
The contangent is the ratio of the cosine and the sine:
The cotangent and tangent are reciprocal to each other: $\cot x = \frac1{\tan x}$.
The cotangent and tangent are complementary to each other: $\cot x = \tan (\frac{\pi}{2} - x)$.
Double angle formula: $\tan x = \cot x - 2\cot 2x$.
The hyperbolic analog $\coth x = \frac{e^x + e^{-x}}{e^x - e^{-x}}$ is related to $\cot x$ via the formula
Meanwhile $\coth x$ is related to the Bernoulli numbers $B_n$, defined by the exponential generating function
through a series of equations
Notice the right side defines an even function. Therefore
and so
The cotangent is the logarithmic derivative? of the sine function:
Applying this observation to the Euler-Weierstrass product formula for the sine function (see there for a proof):
one obtains the following summation formula for the cotangent:
This expansion was used by Eisenstein as a starting point for developing the theory of trigonometric functions; Eisenstein’s account of elliptic functions (cf. the eponymous Eisenstein series), 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.
The power series identity
holds over an open domain where the series converges, ${|x|} \lt 1$.
From the Eisenstein expansion, we have
By a geometric series expansion, the last expression is
which is the same as $1 - 2\sum_{k \geq 1} \zeta(2k)x^{2k}$.
Wikipedia, Trigonometric functions – tan
Veeravalli Varadarajan, Circular Functions (pdf).
Last revised on June 7, 2023 at 10:39:03. See the history of this page for a list of all contributions to it.