Series Representation for sin(x)/(cos(x)+cosh(x)) Valid for 0<x?

[rman144]

Does anyone know of a series representation for:

$$\frac{sin(x)}{cos(x)+cosh(x)}$$

Preferably valid for 0<x, but any ideas or assistance on any domain would be much appreciated.

 

[John Creighto]

Do a taylor series, for sin(x), cos(x), cosh(x), 1/x, then compose them and use the multinational theorem.

 

[zcd]

$$\sin(x)=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}$$
$$\cos(x)=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n}}{(2n)!}=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+...$$
$$\cosh(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\frac{x^{6}}{6!}+...$$

You could cancel some of the terms on denominator to get $$1+\frac{x^{4}}{4!}+\frac{x^{8}}{8!}+...=\sum_{n=0}^{\infty}\frac{x^{4n}}{(4n)!}$$ and then do long division.

Wolfram Alpha gives
$$\frac{x}{2}-\frac{x^{3}}{12}+\frac{x^{5}}{60}-\frac{17x^{7}}{5040}+\frac{31x^{9}}{45360}-\frac{691x^{11}}{4989600}+...$$

 

[g_edgar]

And Maple disagrees in some signs...
$${\frac {\sin \left( x \right) }{\cos \left( x \right) +\cosh \left( x<br /> \right) }} = {\frac {1}{2}}x-{\frac {1}{12}}{x}^{3}-{\frac {1}{60}}{x}^{5}+{\frac <br /> {17}{5040}}{x}^{7}+{\frac {31}{45360}}{x}^{9}-{\frac {691}{4989600}}{x<br /> }^{11}+O \left( {x}^{12} \right)$$