# $\cos\large\frac{\pi}{11}$$\cos\large\frac{2\pi}{11}$$\cos\large\frac{3\pi}{11}..........$$\cos\large\frac{11\pi}{11} is equal to (a)\;-\large\frac{1}{32}$$\qquad(b)\;\large\frac{1}{512}$$\qquad(c)\;\large\frac{1}{1024}$$\qquad(d)\;-\large\frac{1}{2048}$

$\cos\large\frac{\pi}{11}$$\cos\large\frac{2\pi}{11}$$\cos\large\frac{3\pi}{11}..........$$\cos\large\frac{11\pi}{11} \Rightarrow\big [\cos\large\frac{\pi}{11}$$\cos\large\frac{2\pi}{11}$$\cos\large\frac{3\pi}{11}$$\cos\large\frac{4\pi}{11}$$\cos\large\frac{5\pi}{11}\big]^2 \Rightarrow \big[\cos\large\frac{\pi}{11}$$\cos\large\frac{2\pi}{11}$$\cos\large\frac{4\pi}{11}$$\cos\large\frac{8\pi}{11}$$\cos\large\frac{5\pi}{11}\big]^2 \Rightarrow \bigg[\large\frac{\sin 16\Large\frac{\pi}{11}}{16\sin\Large\frac{\pi}{11}}$$\cos\large\frac{5\pi}{11}\bigg]^2$
$\Rightarrow \bigg[\large\frac{2\sin\Large\frac{5\pi}{11}.\cos\Large\frac{5\pi}{11}}{32\sin\Large\frac{\pi}{11}}\bigg]^2$
$\Rightarrow \large\frac{1}{1024}$
Hence (c) is the correct answer.