Example 6. Express $\sin ^{3} x \cos ^{5} x$ as a sum or difference.
Let $\mathrm{z}=\cos \mathrm{x}+\mathrm{i} \sin \mathrm{x}$, by formulas (A) and (B) we get $\sin ^{3} x \cos ^{5} x$
$$
\begin{aligned}
= & \left[\frac{i\left(1-z^{2}\right)}{2 z}\right]^{3}\left(\frac{z^{2}+1}{2 z}\right)^{5} \\
= & -\frac{i}{256 z^{8}}\left(-z^{10}-2 z^{14}+2 z^{12}+6 z^{10}\right. \\
= & -\frac{1}{128}\left[\frac{i\left(1-z^{6}-2 z^{4}+2 z^{2}+1\right)}{2 z^{8}}+\frac{2 i\left(1-z^{12}\right)}{2 z^{8}}\right. \\
& \left.-\frac{2 i\left(1-z^{8}\right)}{2 z^{4}}-\frac{6 i\left(1-z^{4}\right)}{2 z^{2}}\right] \\
= & -\frac{1}{128}(\sin 8 x+2 \sin 6 x-2 \sin 4 x \\
& -6 \sin 2 x) .
\end{aligned}
$$