What is the Taylor series expansion for the function f(x)=[1-cos(x)]/sin(x) ?

1 Answer
Jun 25, 2018

1cosxsinx=2n=1(1)n(22n1)B2n(2n)!x2n1

Explanation:

Use the trigonometric identities:

2sin2α=1cos2α

2sinαcosα=sin2α

Then:

1cosxsinx=2sin2(x2)2sin(x2)cos(x2)=tan(x2)

The McLaurin expansion of the tangent is very difficult to obtain, but you can it from a manual:

tan(x2)=n=1(1)n22n(22n1)B2n(2n)!(x2)2n1

tan(x2)=n=1(1)n22n(22n1)B2n(2n)!22n1x2n1

tan(x2)=2n=1(1)n(22n1)B2n(2n)!x2n1