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$\log \left(\sin 1^{\circ}\right) \cdot \log \left(\sin 2^{\circ}\right) \cdot \log \left(\sin 3^{\circ}\right)$ $\ldots \log \left(\sin 179^{\circ}\right)$
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is zero
$\log \left(\sin 1^{\circ}\right) \cdot \log \left(\sin 2^{\circ}\right) \cdot \log \left(\sin 3^{\circ}\right) \ldots \log \left(\sin 179^{\circ}\right)$
$=\log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log \sin 90^{\circ} \ldots \log \sin 179^{\circ}$
$=\log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log (1) \ldots \log \sin 179^{\circ}$
$\left.=\log \sin 1^{\circ} \cdot \log \sin 2 \ldots 0 \ldots \sin 90^{\circ}=1\right)$
$=0 \quad\left(\because \log \sin 179^{\circ} 1=0\right)$
$=\log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log \sin 90^{\circ} \ldots \log \sin 179^{\circ}$
$=\log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log (1) \ldots \log \sin 179^{\circ}$
$\left.=\log \sin 1^{\circ} \cdot \log \sin 2 \ldots 0 \ldots \sin 90^{\circ}=1\right)$
$=0 \quad\left(\because \log \sin 179^{\circ} 1=0\right)$
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