Editing Trigonometric functions (section)

===Power series expansion===
The basic trigonometric functions can be defined by the following power series expansions.<ref>Whitaker and Watson, p 584</ref> These series are also known as the [[Taylor series]] or [[Maclaurin series]] of these trigonometric functions:
:<math>
\begin{align}
\sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[6mu]
& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\[8pt]
\cos x & = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\[6mu]
& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}.
\end{align}
</math>
The [[radius of convergence]] of these series is infinite. Therefore, the sine and the cosine can be extended to [[entire function]]s (also called "sine" and "cosine"), which are (by definition) [[complex-valued function]]s that are defined and [[holomorphic]] on the whole [[complex plane]].

Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.

Being defined as fractions of entire functions, the other trigonometric functions may be extended to [[meromorphic function]]s, that is functions that are holomorphic in the whole complex plane, except some isolated points called [[zeros and poles|poles]]. Here, the poles are the numbers of the form <math display="inline">(2k+1)\frac \pi 2</math> for the tangent and the secant, or <math>k\pi</math> for the cotangent and the cosecant, where {{mvar|k}} is an arbitrary integer.

Recurrences relations may also be computed for the coefficients of the [[Taylor series]] of the other trigonometric functions. These series have a finite [[radius of convergence]]. Their coefficients have a [[combinatorics|combinatorial]] interpretation: they enumerate [[alternating permutation]]s of finite sets.<ref>Stanley, Enumerative Combinatorics, Vol I., p. 149</ref>

More precisely, defining
: {{mvar|U<sub>n</sub>}}, the {{mvar|n}}th [[up/down number]],
: {{mvar|B<sub>n</sub>}}, the {{mvar|n}}th [[Bernoulli number]], and
: {{mvar|E<sub>n</sub>}}, is the {{mvar|n}}th [[Euler number]],
one has the following series expansions:<ref>Abramowitz; Weisstein.</ref> 
: <math>
\begin{align}
\tan x & {} = \sum_{n=0}^\infty \frac{U_{2n+1}}{(2n+1)!}x^{2n+1} \\[8mu]
& {} = \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} \left(2^{2n}-1\right) B_{2n}}{(2n)!}x^{2n-1} \\[5mu]
& {} = x + \frac{1}{3}x^3 + \frac{2}{15}x^5 + \frac{17}{315}x^7 + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
\end{align}
</math>

: <math>
\begin{align}
\csc x &= \sum_{n=0}^\infty \frac{(-1)^{n+1} 2 \left(2^{2n-1}-1\right) B_{2n}}{(2n)!}x^{2n-1} \\[5mu]
&= x^{-1} + \frac{1}{6}x + \frac{7}{360}x^3 + \frac{31}{15120}x^5 + \cdots, \qquad \text{for } 0 < |x| < \pi.
\end{align}
</math>

: <math>
\begin{align}
\sec x &= \sum_{n=0}^\infty \frac{U_{2n}}{(2n)!}x^{2n}
= \sum_{n=0}^\infty \frac{(-1)^n E_{2n}}{(2n)!}x^{2n} \\[5mu]
&= 1 + \frac{1}{2}x^2 + \frac{5}{24}x^4 + \frac{61}{720}x^6 + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
\end{align}
</math>

: <math>
\begin{align}
\cot x &= \sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n}}{(2n)!}x^{2n-1} \\[5mu]
&= x^{-1} - \frac{1}{3}x - \frac{1}{45}x^3 - \frac{2}{945}x^5 - \cdots, \qquad \text{for } 0 < |x| < \pi.
\end{align}
</math>