Question

$\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{16\pi }{15}=$

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $\frac{1}{16}$

Answer 1

The correct option is D

$\frac{1}{16}$

Explanation for the correct option:

Step 1. Multiply and divide the given expression by $2\sin \frac{2\mathrm{\pi }}{15}$, we get

$\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{16\pi }{15}$

$=\frac{1}{2\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left(2\sin \frac{2\mathrm{\pi }}{15}\times \cos \frac{2\mathrm{\pi }}{15}\right)\times \cos \frac{4\mathrm{\pi }}{15}\times \cos \frac{8\mathrm{\pi }}{15}\times \cos \frac{16\mathrm{\pi }}{15}$

$=\frac{1}{2\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left(\sin \frac{4\mathrm{\pi }}{15}\right)\times \cos \frac{4\mathrm{\pi }}{15}\times \cos \frac{8\mathrm{\pi }}{15}\times \cos \frac{16\mathrm{\pi }}{15}$ ; $\left[\mathbf{\because }\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\mathit{A}\mathit{c}\mathit{o}\mathit{s}\mathit{A}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{2}\mathit{A}\right]$

Step 2. Multiple and divide it by 2,

$=\frac{1}{2\times 2\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left(2\sin \frac{4\mathrm{\pi }}{15}\times \cos \frac{4\mathrm{\pi }}{15}\right)\times \cos \frac{8\mathrm{\pi }}{15}\times \cos \frac{16\mathrm{\pi }}{15}$

$=\frac{1}{4\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left(\sin \frac{8\mathrm{\pi }}{15}\right)\times \cos \frac{8\mathrm{\pi }}{15}\times \cos \frac{16\mathrm{\pi }}{15}$

Step 3. Again Multiply and divide it by 2,

$=\frac{1}{2\times 4\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left(2\sin \frac{8\mathrm{\pi }}{15}\times \cos \frac{8\mathrm{\pi }}{15}\right)\times \cos \frac{16\mathrm{\pi }}{15}$

$=\frac{1}{8\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left(\sin \frac{16\mathrm{\pi }}{15}\right)\times \cos \frac{16\mathrm{\pi }}{15}$

Step 4. Again multiply and divide it by 2,

$=\frac{1}{2\times 8\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left(2\sin \frac{16\mathrm{\pi }}{15}\times \cos \frac{16\mathrm{\pi }}{15}\right)$

$=\frac{1}{16\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left(2\sin \frac{16\mathrm{\pi }}{15}\times \cos \frac{16\mathrm{\pi }}{15}\right)$

$=\frac{1}{16\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left(\sin \frac{32\mathrm{\pi }}{15}\right)$

$=\frac{1}{16\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \left[\sin \left(2\pi +\frac{2\mathrm{\pi }}{15}\right)\right]$ ; $\left[\mathbf{\because }\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{2}\mathbf{\pi }\mathbf{+}\mathbf{\theta }\mathbf{)}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\right]$

$=\frac{1}{16\sin {\displaystyle \frac{2\mathrm{\pi }}{15}}}\times \sin \left(\frac{2\mathrm{\pi }}{15}\right)$

$=\frac{\mathbf{1}}{\mathbf{16}}$

Hence, Option ‘D’ is Correct.