\(\large \sin10^\circ \cdot \sin30^\circ \cdot \sin50^\circ \cdot \sin70^\circ\color{blue}= 0.0625 \)
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\({\displaystyle {sin\ x=\frac {1}{2\mathrm {i} }}\left(\mathrm {e} ^{\mathrm {i} x}-\mathrm {e} ^{-\mathrm {i} x}\right)}\)
\(\large \sin\frac{\pi}{18} \cdot \sin\frac{3\pi}{18} \cdot \sin\frac{5\pi}{18} \cdot \sin\frac{7\pi}{18}\)
\(={\displaystyle {\frac {1}{2\mathrm {i} }}\left(\mathrm {e} ^{\mathrm {i} \frac{\pi}{18}}-\mathrm {e} ^{-\mathrm {i} \frac{\pi}{18}}\right)}\times\)\({\displaystyle {\frac {1}{2\mathrm {i} }}\left(\mathrm {e} ^{\mathrm {i} \frac{3\pi}{18}}-\mathrm {e} ^{-\mathrm {i} \frac{3\pi}{18}}\right)}\)
\(\times {\displaystyle {\frac {1}{2\mathrm {i} }}\left(\mathrm {e} ^{\mathrm {i} \frac{5\pi}{18}}-\mathrm {e} ^{-\mathrm {i} \frac{5\pi}{18}}\right)}\)\({\displaystyle {\times \frac {1}{2\mathrm {i} }}\left(\mathrm {e} ^{\mathrm {i} \frac{7\pi}{18}}-\mathrm {e} ^{-\mathrm {i} \frac{7\pi}{18}}\right)}\)
\(\large =0.0625\)
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