Is there a quick way to find the answer? Obviously the Taylor approximation is exact at the point where it is centered ($\sin(x) = x$, for $x = 0$) but where else is the approximation 100% valid?
For example, what values of $x$ exactly satisfy the following relation $$\sin(x) = x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5$$
It won't be many as the absolute value of that expression will exceed $1$ after a short time. For the few solutions within that range, an analytical solution is probably not possible and you will need to solve it by numerical means.
If want equality everywhere to a truncated Taylor series then there is a name of the class of function that satisfies this: a polynomial.