A) \[\frac{1}{16}\]
B) \[\frac{3}{64}\]
C) \[\frac{3}{15}\]
D) \[0\]
Correct Answer: A
Solution :
\[\sin 12{}^\circ \sin 24{}^\circ sin48{}^\circ sin84{}^\circ \] \[=\frac{1}{4}[2\sin 12{}^\circ \sin 48{}^\circ ][2\sin 24{}^\circ \sin 84{}^\circ ]\] \[=\frac{1}{4}[\cos 36{}^\circ -\cos 60{}^\circ ][\cos 60\cos 108{}^\circ ]\] \[=\frac{1}{4}\left[ \cos 36{}^\circ -\frac{1}{2} \right]\left[ \frac{1}{2}+\sin 18{}^\circ \right]\] \[=\frac{1}{4}\left[ \frac{\sqrt{5}+1}{4} \right]\left[ \frac{1}{2}+\frac{\sqrt{5}+1}{4} \right]\] \[=\frac{1}{64}(\sqrt{5}-1)(\sqrt{5}+1)=\frac{1}{16}\]
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