Question

Solve $cos\frac{\pi }{11}cos\frac{2\pi }{11}cos\frac{3\pi }{11}cos\frac{4\pi }{11}cos\frac{5\pi }{11}$

• A. $\frac{1}{4}$
• B. $\frac{1}{8}$
• C. $\frac{1}{16}$
• D. $\frac{1}{32}$

Answer 1

The correct option is D $\frac{1}{32}$

$\Rightarrow \cos \frac{\pi }{11}\cos \frac{2\pi }{11}\cos \frac{3\pi }{11}\cos \frac{4\pi }{11}\cos \frac{5\pi }{11}$

$\Rightarrow \frac{2\sin \frac{\pi }{11}\cos \frac{\pi }{11}\cos \frac{2\pi }{11}\cos \frac{3\pi }{11}\cos \frac{4\pi }{11}\cos \frac{5\pi }{11}}{2\sin \frac{\pi }{11}}$

$\Rightarrow \{\because \sin 2\theta =2\sin \theta \cos \theta \}$

$\Rightarrow \frac{\sin \frac{2\pi }{11}\cos \frac{2\pi }{11}\cos \frac{3\pi }{11}\cos \frac{4\pi }{11}\cos \frac{5\pi }{11}}{2\sin \frac{\pi }{11}}$

$=\frac{\frac{1}{2}\sin \frac{4\pi }{11}\cos \frac{3\pi }{11}\cos \frac{4\pi }{11}\cos \frac{5\pi }{11}}{2\sin \frac{\pi }{11}}$

$=\frac{\frac{1}{4}\sin \frac{8\pi }{11}\cos \frac{3\pi }{11}\cos \frac{5\pi }{11}}{2\sin \frac{\pi }{11}}$

$=\frac{\frac{1}{4}\sin (\pi -\frac{3\pi }{11})\cos \left(\frac{3\pi }{11}\right)\cos \frac{5\pi }{11}}{2\sin \frac{\pi }{11}}$

$=\frac{\frac{1}{4}\sin \frac{3\pi }{11}\cos \frac{3\pi }{11}\cos \frac{5\pi }{11}}{2\sin \frac{\pi }{11}}$

$=\frac{\frac{1}{8}\sin \frac{6\pi }{11}\cos \frac{5\pi }{11}}{2\sin \frac{\pi }{11}}$

$=\frac{\frac{1}{8}\sin (\pi -\frac{5\pi }{11})\cos \frac{5\pi }{11}}{2\sin \frac{\pi }{11}}$

$=\frac{\frac{1}{8}\sin \frac{5\pi }{11}\cos \frac{5\pi }{11}}{2\sin \frac{\pi }{11}}=\frac{\frac{1}{16}\sin \frac{10\pi }{11}}{2\sin \frac{\pi }{11}}=\frac{\frac{1}{16}\sin (\pi -\frac{\pi }{11})}{2\sin \frac{\pi }{11}}$

$=\frac{\frac{1}{16}\sin \left(\frac{\pi }{11}\right)}{2\sin \left(\frac{\pi }{11}\right)}=\frac{1}{32}$