A) \[\frac{1}{2}\]
B) \[\frac{1}{4}\]
C) \[\frac{1}{8}\]
D) \[\frac{1}{16}\]
Correct Answer: C
Solution :
| \[\sin 10{}^\circ \sin 50{}^\circ \sin 70{}^\circ \] |
| \[=\frac{1}{2}\sin 10{}^\circ \,\,(2\sin 70{}^\circ \sin 50{}^\circ )\] |
| \[=\frac{1}{2}\sin 10{}^\circ [\cos (70{}^\circ -50{}^\circ )-\cos (70{}^\circ +50{}^\circ )]\] |
| \[=\frac{1}{2}\sin 10{}^\circ [\cos 20{}^\circ -\cos 120{}^\circ ]\] |
| \[=\frac{1}{2}\sin 10{}^\circ \left[ \cos 20{}^\circ +\frac{1}{2} \right]\] |
| \[=\frac{1}{2}\sin 10{}^\circ \cos 20{}^\circ +\frac{1}{4}\sin 10{}^\circ \] |
| \[=\frac{1}{4}[\sin (10{}^\circ +20{}^\circ )+\sin \,\,(10{}^\circ -20{}^\circ )]+\frac{1}{4}\sin 10{}^\circ \] |
| \[=\frac{1}{4}\sin 30{}^\circ -\frac{1}{4}\sin 10{}^\circ +\frac{1}{4}\sin 10{}^\circ =\frac{1}{4}\cdot \frac{1}{2}=\frac{1}{8}\] |
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