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The value of cos π 65 cos 2 π 65 cos 4 π 65 cos 8 π 65 cos 16 π 65 cos 32 π 65 is - Mathematics

MCQ

The value of \[\cos \frac{\pi}{65} \cos \frac{2\pi}{65} \cos \frac{4\pi}{65} \cos \frac{8\pi}{65} \cos \frac{16\pi}{65} \cos \frac{32\pi}{65}\]  is 

  

Options

  • \[\frac{1}{8}\]

     

  • \[\frac{1}{16}\]

     

  • \[\frac{1}{32}\]

     

  •  none of these

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Solution

 none of these

\[\text{ We have } , \]
\[\cos\frac{\pi}{65} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ = \frac{2\sin\frac{\pi}{65}}{2\sin\frac{\pi}{65}} \cos\frac{\pi}{65} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ \left( \text{ dividing and multiplying by }  2\sin\frac{\pi}{65} \right)\]
\[ = \frac{2\sin\frac{2\pi}{65}}{2 \times 2\sin\frac{\pi}{65}} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ = \frac{2\sin\frac{4\pi}{65}}{2 \times 4\sin\frac{\pi}{65}} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]

\[= \frac{2\sin\frac{8\pi}{65}}{2 \times 8\sin\frac{\pi}{65}} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ = \frac{2\sin\frac{16\pi}{65}}{2 \times 16\sin\frac{\pi}{65}} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}\]
\[ = \frac{2\sin\frac{32\pi}{65}}{2 \times 32\sin\frac{\pi}{65}} \cos\frac{32\pi}{65}\]
\[ = \frac{\sin\frac{64\pi}{65}}{64\sin\frac{\pi}{65}}\]
\[ = \frac{\sin\left( \pi - \frac{\pi}{65} \right)}{64\sin\frac{\pi}{65}}\]
\[ = \frac{\sin\frac{\pi}{65}}{64\sin\frac{\pi}{65}}\]
\[ = \frac{1}{64}\]

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Q 3 | Page 43
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