A) \[\frac{1}{256}\]
B) \[\frac{1}{2}\]
C) \[\frac{1}{1024}\]
D) \[\frac{1}{512}\]
Correct Answer: D
Solution :
\[\left\{ \cos \,\theta .\cos ({{2}^{1}}\theta ).cos({{2}^{2}}\theta ).....\,\cos ({{2}^{n-1}}\theta )\,=\,\frac{\sin {{2}^{n}}\theta }{{{2}^{n}}\,\sin \,\theta } \right\}\] \[=\,\,\left( \cos \frac{\pi }{{{2}^{2}}}.\cos \frac{\pi }{{{2}^{3}}}.\cos \frac{\pi }{{{2}^{4}}}\,....\,\cos \,\frac{\pi }{{{2}^{10}}} \right)\,\,\sin \,\,\frac{\pi }{{{2}^{10}}}\] \[=\,\,\left( \cos \frac{\pi }{{{2}^{10}}}.\cos \frac{\pi }{{{2}^{9}}}.\cos \frac{\pi }{{{2}^{8}}}\,....\,\cos \,\frac{\pi }{{{2}^{2}}} \right)\,\,\sin \left( \frac{\pi }{{{2}^{10}}} \right)\] \[=\frac{\sin \left( {{2}^{9}}.\frac{\pi }{{{2}^{10}}} \right)}{{{2}^{9}}\,\sin \left( \frac{\pi }{{{2}^{10}}} \right)}\,.\,\sin \,\frac{\pi }{{{2}^{10}}}\] \[=\,\,\,\frac{\sin \,\left( \frac{\pi }{2} \right)}{{{2}^{9}}}\,\,=\,\,\frac{1}{512}\]
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