Can I consider: $\cos(x)=(1+a_0)(1+a_1x)(1+a_2x^2)\cdots$?
Taking the logarithm on both sides and differentiating gives: ($a_0$ considered to be $0$)
$-\tan(x)=\dfrac{a_1}{a_1x+1}+\dfrac{2a_2x}{a_2x^2+1}+\cdots; a_1=0$
$-\sec^2(x)=\dfrac{-2a_2(a_2x^2-1)}{(a_2x^2+1)^2}+\cdots; a_2=\dfrac{-1}{2}$
And so on,
$a_2=\dfrac{-1}{2},a_2=\dfrac{1}{12},a_2=\dfrac{-1}{45},\cdots$
$\cos(x)=(1-\dfrac{1\cdot x^2}{2!})\cdot(1+\dfrac{2\cdot x^4}{4!})\cdot(1-\dfrac{16\cdot x^6}{6!})\cdot(1+\dfrac{272\cdot x^8}{8!})\cdots$, Where $1,2,16,272,7936,353792,\cdots$ are "Zag numbers" https://oeis.org/A000182
Is this series representation wrong? If so, why?