Start by taking consecutive derivatives of \(\displaystyle sec(x)\); \(\displaystyle f'(x), f''(x), f'''(x), ...., f^{n}(x)\)
Using c=0 will give you the MacLaurin series.
\(\displaystyle \L\\\sum_{k=0}^{\infty}\frac{f^{k}(0)}{k!}x^{k}=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}+............+\frac{f^{k}(0)}{k!}x^{k}+......\)
I'll do a few.
\(\displaystyle sec(0)=1\)
\(\displaystyle f'(0)=sec(x)tan(x)=sec(0)tan(0)=0\)
\(\displaystyle f''(0)=sec^{3}(x)+tan^{2}(x)sec(x)=sec^{3}(0)+tan^{2}(0)sec(0)=1\)
Odd derivatives will give 0. It looks like you'll need the 2nd, 4th, 6th, and 8th derivatives.
Use the formula with your derivative results.
