Question

The value of $cos\frac{\pi }{11}cos\frac{2\pi }{11}cos\frac{3\pi }{11}\dots \dots cos\frac{11\pi }{11}$ is equal to

• A. $-\frac{1}{32}$
• B. $\frac{1}{512}$
• C. $\frac{1}{1024}$
• D. $-\frac{1}{2048}$

Answer 1

Answer: (C)

Required expression $={\left(cos\frac{\pi }{11}cos\frac{2\pi }{11}cos\frac{3\pi }{11}cos\frac{4\pi }{11}cos\frac{5\pi }{11}\right)}^2$
$={\left(cos\frac{\pi }{11}cos\frac{2\pi }{11}cos\frac{8\pi }{11}cos\frac{4\pi }{11}cos\frac{5\pi }{11}\right)}^2$
$={\left(\frac{sin\frac{16\pi }{11}}{16sin\frac{\pi }{11}}\times cos\frac{5\pi }{11}\right)}^2$
$={\left(\frac{2sin\frac{5\pi }{11}cos\frac{5\pi }{11}}{32sin\frac{\pi }{11}}\right)}^2$
$={\left(\frac{sin\frac{10\pi }{11}}{32\times sin\frac{\pi }{11}}\right)}^2=\frac{1}{{\left(32\right)}^2}=\frac{1}{1024}$