# Thread: Taylor expansion of composite function

1. ## Taylor expansion of composite function

Hi,

If i want to use Taylor expansion on a function like:

$\displaystyle \frac{ln(sin(x))}{cosx}$

Do i have to substitute the expansion of sin x, in to the expansion of ln x?

2. Originally Posted by Jones
Hi,

If i want to use Taylor expansion on a function like:

$\displaystyle \frac{ln(sin(x))}{cosx}$

Do i have to substitute the expansion of sin x, in to the expansion of ln x?
Yes, that would be a good idea. Then divide everything by the expansion for $\displaystyle \cos{x}$.

3. Originally Posted by Jones
Hi,

If i want to use Taylor expansion on a function like:

$\displaystyle \frac{ln(sin(x))}{cosx}$

Do i have to substitute the expansion of sin x, in to the expansion of ln x?
For certain interval restrictions $\displaystyle \frac{\ln(\sin(x))}{\cos(x)}=\frac{\ln\left(\sqrt{ 1-\cos^2(x)}\right)}{\cos(x)}=\frac{\ln\left(1-\cos^2(x)\right)}{2\cos(x)}$ and since $\displaystyle \left|\cos^2(x)\right|<1$ (with the proper interval restrictions) we see that $\displaystyle \frac{\ln\left(1-\cos^2(x)\right)}{\cos(x)}=\frac{1}{\cos(x)}\sum_{ n=1}^{\infty}\frac{\cos^{2n}(x)}{n}=\sum_{n=1}^{\i nfty}\frac{\cos^{2n-1}(x)}{n}$