Q. $\sin 12^{\circ} \cdot \sin 24^{\circ} \cdot \sin 48^{\circ} \cdot \sin 84^{\circ}=$

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A

$1 / 64$

B

$1 / 16$

C

$3 / 16$

D

$5 / 32$

Solution:

$\sin 12^{\circ} \cdot \sin 24^{\circ} \cdot \sin 48^{\circ} \cdot \sin 84^{\circ}$
$=\frac{1}{\sin 72^{\circ} \cdot \sin 36^{\circ}}\left(\sin 12^{\circ} \cdot \sin 72^{\circ} \cdot \sin 48^{\circ}\right) \cdot\left(\sin 24^{\circ} \cdot \sin 36^{\circ} \cdot \sin 84^{\circ}\right) $
$=\frac{1}{\sin 72^{\circ} \cdot \sin 36^{\circ}} \cdot \frac{1}{4}\left(\sin 3\left(12^{\circ}\right)\right) \cdot \frac{1}{4} \cdot\left(\sin 3\left(24^{\circ}\right)\right)=\frac{1}{16}$
( Note $ \sin \theta \cdot \sin \left(60^{\circ}+\theta\right) \cdot \sin \left(60^{\circ}-\theta\right)=\frac{1}{4} \cdot \sin 3 \theta$ )

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