# Given 1- cosx, how do you find the Taylor polynomial?

##### 1 Answer

Mar 11, 2017

We should know that:

#cosx=sum_(n=0)^oo(-1)^nx^(2n)/((2n)!)=1-x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)+...#

So

#1-cosx=1-(1-x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)+...)#

#color(white)(1-cosx)=x^2/(2!)-x^4/(4!)+x^6/(6!)-x^8/(8!)+...#

Noting this still alternates and that the factorials and powers increase by

#1-cosx=sum_(n=0)^oo(-1)^nx^(2n+2)/((2n+2)!)#

Which we could also write as:

#1-cosx=sum_(n=1)^oo(-1)^(n-1)x^(2n)/((2n)!)#