= Solution
Take the <imaginary part> of the eighth-power binomial expansion in <De Moivre's theorem>. With $s=\sin\theta$ and $c=\cos\theta$,
$$
\sin8\theta=8c^7s-56c^5s^3+56c^3s^5-8cs^7
=sc\left[8c^6-56c^4s^2+56c^2s^4-8s^6\right].
$$
Substitute $c^2=1-s^2$ and collect powers:
$$
\sin8\theta=sc\left[8-80s^2+192s^4-128s^6\right].
$$
Therefore the requested leading coefficient is
$$
\boxed{a=-128.}
$$
The other coefficients, if retained as an algebraic check, are $b=192,c=-80,d=8$. No division by $s$ or $c$ is needed, so the identity remains valid when either vanishes.
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