Originally Posted by

**CaptainBlack** $\displaystyle

\sec(x)=\frac{1}{\cos(x)}=\frac{1}{1-x^2/2!+x^4/4!-x^6/6!+...}

$$\displaystyle

=\frac{1}{\cos(x)}=\frac{1}{1- (x^2/2!-x^4/4!+x^6/6!-...) }

$

............. $\displaystyle =1+(x^2/2!-x^4/4!+x^6/6!-...) +(x^2/2!-x^4/4!+x^6/6!-...)^2 $

................... $\displaystyle +(x^2/2!-x^4/4!+x^6/6!-...)^3+..$

The first four non-zero terms will be the $\displaystyle 0,\ 2,\ 4,\ 6$ th powers of $\displaystyle x$

So:

$\displaystyle

\sec(x) = 1 +(x^2/2!-x^4/4!+x^6/6!)+[x^4/(2!)^2-2x^6/(2!4!)]+[x^6/(2!)^3]+O(x^8)

$

RonL