Lets say we have a polynomial $P_n(x)≈\cos(x)$ and assume $P_n(x)=a_0+a_1x+a_2x^2+...+a_nx^n$ and my proposal is that as $n$ approaches infinity, $$P_\infty(x)=\cos(x)=a_0+a_1x+a_2x^2+...$$
and now, since the polynomial and the function are equal, that must mean that their values at a specific point, lets say $x=0$,for example, are equal too:
$$P_\infty(0)=\cos(0)$$
ant note, that
$$P_\infty(0)=a_0=\cos(0)=0$$
Also since $P_\infty(x)=\cos(x)$ their derivatives are equal too,
$$P'_\infty(x)=a_1+2a_2x+3a_3x^2...=\cos'(x)=-\sin(x)$$
(this uses the fact that the derivative of a constant is zero, and the power rule: $(x^r)'=rx^{r-1}$)
Now lets analize the derivatives at zero
$$P'_\infty(0)=a_1=-sin(0)=0$$
Now the double derivative
$$P''_\infty(x)=-sin'(x)=-cos(x)=2a_2+2*3a_3x+2*3*4a_4x^2...=2!a_2+3!a_3x+4!a_4x^2...$$
(as you can see this is where the factorial comes in, it is merely a consequance of the power rule)
Once again let us analize the double derivative at $0$
$$P''_\infty(0)=2!a_2=-cos(0)=-1$$
$$a_2=-\frac{1}{2!}$$
now, by derriving the funtion ad infinitum and analyzing its value at zero, we may find all of the infinite polynomials coefficients, and by noting that
$$\cos(x)'=-\sin(x)$$
$$\cos(x)''=-\sin(x)'=-\cos(x)$$
$$\cos(x)'''=-\sin(x)''=-\cos(x)'=\sin(x)$$
$$\cos(x)''''=-\sin(x)'''=-\cos(x)''=\sin(x)'=\cos(x)$$
the derivatives of the cosine are periodic with a period of four,
we may note that
$$a_0=1$$
$$a_1=0$$
$$a_2=-1/2!$$
$$a_3=0$$
$$a_4=1/4!$$
$$a_5=0$$
$$a_6=-1/6!$$
or generally, if $n$ is an odd number, $a_n=0$, if $n$ is divisible by $4$, then $a_n=1/n!$, and if $n$ divisible by 2 but not by 4, $a_n=-1/n!$.
Putting that all together, we get that
$$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}...$$
Now as you can see, analyzing the function anywhere else other than at$x=0$ would be a pain, because if, for example $$P_2(x)=a_0+a_1x+a_2x^2$$, and we analyze at $x=3$, we get that $$P_2(3)=a_0+3a_1+9a_2≈\cos(3)$$
So as you can see, getting any coefficients out of this expression would be impossible, further more if we have an infinite polynomial.