Prove that: \[ \sin \frac{13 \pi}{3} \sin \frac{8 \pi}{3}+\cos \frac{2 \pi}{3} \sin \frac{5 \pi}...

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Prove that:
\[
\sin \frac{13 \pi}{3} \sin \frac{8 \pi}{3}+\cos \frac{2 \pi}{3} \sin \frac{5 \pi}{6}=\frac{1}{2}
\]
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