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 equal to

• A. $\frac{1}{64}$
• B. $\frac{-1}{64}$
• C. $\frac{1}{32}$
• D. $\frac{-1}{8}$

Answer 1

The correct option is A $\frac{1}{64}$

Taking $A=\frac{\pi }{65}=\frac{\pi }{2^6+1}$
$\cos \frac{\pi }{65}\cos \frac{2\pi }{65}\cos \frac{4\pi }{65}\cos \frac{8\pi }{65}\cos \frac{16\pi }{65}\cos \frac{32\pi }{65}\cos A\cos 2A\cos 2^{2}A\cos 2^{3}A\cos 2^{4}A\cos 2^{5}A=\frac{1}{2^6}=\frac{1}{64}$