The value of cos

Question:

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

(a) $\frac{1}{8}$

(b) $\frac{1}{16}$

(c) $\frac{1}{32}$

(d) none of these

Solution:

(d) none of these

We have,

$\cos \frac{\pi}{65} \cos \frac{2 \pi}{65} \cos \frac{4 \pi}{65} \cos \frac{8 \pi}{65} \cos \frac{16 \pi}{65} \cos \frac{32 \pi}{65}$

$=\frac{2 \sin \frac{\pi}{65}}{2 \sin \frac{\pi}{65}} \cos \frac{\pi}{65} \cos \frac{2 \pi}{65} \cos \frac{4 \pi}{65} \cos \frac{8 \pi}{65} \cos \frac{16 \pi}{65} \cos \frac{32 \pi}{65}$

(dividing and multiplying by $2 \sin \frac{\pi}{65}$ )

$=\frac{2 \sin \frac{2 \pi}{65}}{2 \times 2 \sin \frac{\pi}{65}} \cos \frac{2 \pi}{65} \cos \frac{4 \pi}{65} \cos \frac{8 \pi}{65} \cos \frac{16 \pi}{65} \cos \frac{32 \pi}{65}$

$=\frac{2 \sin \frac{4 \pi}{65}}{2 \times 4 \sin \frac{\pi}{65}} \cos \frac{4 \pi}{65} \cos \frac{8 \pi}{65} \cos \frac{16 \pi}{65} \cos \frac{32 \pi}{65}$

$=\frac{2 \sin \frac{8 \pi}{65}}{2 \times 8 \sin \frac{\pi}{65}} \cos \frac{8 \pi}{65} \cos \frac{16 \pi}{65} \cos \frac{32 \pi}{65}$

$=\frac{2 \sin \frac{16 \pi}{65}}{2 \times 16 \sin \frac{\pi}{65}} \cos \frac{16 \pi}{65} \cos \frac{32 \pi}{65}$

$=\frac{2 \sin \frac{32 \pi}{65}}{2 \times 32 \sin \frac{\pi}{65}} \cos \frac{32 \pi}{65}$

$=\frac{\sin \frac{64 \pi}{65}}{64 \sin \frac{\pi}{65}}$

$=\frac{\sin \left(\pi-\frac{\pi}{65}\right)}{64 \sin \frac{\pi}{65}}$

$=\frac{\sin \frac{\pi}{65}}{64 \sin \frac{\pi}{65}}$

$=\frac{1}{64}$

 

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