Prove that: \[ \sin \frac{5 \pi}{18}-\cos \frac{4 \pi}{9}=\sqrt{3} \sin \frac{\pi}{9} \]

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Prove that:
\[
\sin \frac{5 \pi}{18}-\cos \frac{4 \pi}{9}=\sqrt{3} \sin \frac{\pi}{9}
\]
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