A) \[\frac{1}{16}\]
B) \[\frac{1}{32}\]
C) \[\frac{1}{64}\]
D) \[\frac{1}{8}\]
Correct Answer: A
Solution :
\[\cos \frac{2\pi }{15}\cdot \cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{16\pi }{15}\] \[=\frac{1}{2\sin \frac{\pi }{15}}\cdot 2\sin \frac{2\pi }{15}\cdot \cos \frac{2\pi }{15}\cdot \cos \frac{4\pi }{15}\cdot \cos \frac{8\pi }{15}\]\[\cos \frac{16\pi }{15}\] \[\Rightarrow \]\[\frac{1}{2\sin \frac{2\pi }{15}}\cdot \sin \frac{4\pi }{15}\cdot \cos \frac{4\pi }{15}\cdot \cos \frac{8\pi }{15}\cdot \cos \frac{16\pi }{15}\] \[\Rightarrow \]\[\frac{1}{4\sin \frac{2\pi }{15}}\cdot \sin \frac{8\pi }{15}\cdot \cos \frac{8\pi }{15}\cdot \cos \frac{16\pi }{15}\] \[=\frac{1}{8\sin \frac{2\pi }{15}}\cdot \sin \frac{16\pi }{15}\cdot \cos \frac{16\pi }{15}\] \[=\frac{1}{16\sin \frac{2\pi }{15}}.\sin \frac{32\pi }{15}\] \[=\frac{1}{16\sin \frac{2\pi }{15}}\cdot \sin \left( 2\pi +\frac{2\pi }{15} \right)=\frac{1}{16\sin \frac{2\pi }{15}}\cdot \sin \frac{2\pi }{15}\] \[=\frac{1}{16}\]
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