91Ó°ÊÓ

The given function.

$f\left(x\right)=\cos  x$

The Taylor’s series is a power series that yields the expansion of a function in the vicinity of a point if the function is continuous, all of its derivatives exist, and the series converges.

Rewrite $\cos x$and use $\cos (\mathrm{Ï€}+x)=-\cos (x)$,

$$\begin{gathered} \cos \left(x\right)=\cos [\mathrm{Ï€}+(x-\mathrm{Ï€}\left)\right] \\ \cos [\mathrm{Ï€}+(x-\mathrm{Ï€}\left)\right]=-\cos (x-\mathrm{Ï€}) \end{gathered}$$

Use the Maclaurin series of $\cos x$and replace $x$by $(x-\mathrm{Ï€})$,

$\cos \left(x\right)=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+…$

$-\cos \left(x\right)=-\left[1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+…\right]$

$-\cos \left(x\right)=-1+\frac{x^2}{2}-\frac{x^4}{24}+\frac{x^6}{720}-…$

$-\cos \left(x\right)=-1+\frac{(x-\mathrm{π}{)}^2}{2}-\frac{(x-\mathrm{π}{)}^4}{24}+\frac{(x-\mathrm{π}{)}^6}{720}-…$

Write first few terms of Taylor’s series aboutlocalid="1657417586942" $a-\mathrm{π}.$

localid="1657417074607" $\cos \left(x\right)=-1+\frac{(x-\mathrm{π}{)}^2}{2}-\frac{(x-\mathrm{π}{)}^4}{24}+\frac{(x-\mathrm{π}{)}^6}{720}-…$

Hence, first few terms of Taylor’s series expansion are:

$\cos \left(x\right)=-1+\frac{(x-\mathrm{π}{)}^2}{2}-\frac{(x-\mathrm{π}{)}^4}{24}+\frac{(x-\mathrm{π}{)}^6}{720}-…$