I'm trying to solve a problem from a previous exam. Unfortunately there is no solution for this problem.
So, the problem is:
Calculate the Taylor polynommial (degree $4$) in $x_0 = 0$ of the function: $f(x) = \frac{1}{1+\cos(x)}$
What I tried so far:
- calculate all $4$ derivatives
- $1+\cos(x) = 2\cos^2(\frac{x}{2})$ and work with this formula
- $\int\frac{1}{1+\cos(x)}dx = \tan(\frac{x}{2})$ and then use the Taylor series of $\tan(\frac{x}{2})$
- $\frac{1}{1 + \cos(x)} = \frac{1}{1 + \left(1 + \frac{x^2}{2!} + \cdots\right)}$
What do you think, is there a good way to calculate the Taylor polynomial of this function or is there just the hard way (derivatves)?