The value of the product cos⁡π15cos⁡2π15cos⁡3π15…cos⁡14π15 is

Let P=cos⁡π15cos⁡2π15cos⁡3π15…cos⁡14π15=−cos⁡π15cos⁡2π15cos⁡4π15cos⁡8π152.cos⁡3π15cos⁡5π15cos⁡6π152=−sin⁡16π1524⋅sin⁡π152⋅5+14⋅12⋅5−142=−1214

The value of the product cos⁡π15cos⁡2π15cos⁡3π15…cos⁡14π15 is

Let P=cos⁡π15cos⁡2π15cos⁡3π15…cos⁡14π15=−cos⁡π15cos⁡2π15cos⁡4π15cos⁡8π152.cos⁡3π15cos⁡5π15cos⁡6π152=−sin⁡16π1524⋅sin⁡π152⋅5+14⋅12⋅5−142=−1214

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The value of the product $\cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \dots \cos \frac{14 π}{15}$ is

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$- \frac{1}{2^{10}}$

$- \frac{1}{2^{14}}$

$- \frac{1}{2^{12}}$

$\frac{1}{2^{14}}$

answer is B.

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Detailed Solution

Let $P = \cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \dots \cos \frac{14 π}{15}$

$= - {(\cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15})}^{2} . {(\cos \frac{3 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15})}^{2}$

$= - {(\frac{\sin \frac{16 π}{15}}{2^{4} \cdot \sin \frac{π}{15}})}^{2} \cdot {(\frac{\sqrt{5} + 1}{4} \cdot \frac{1}{2} \cdot \frac{\sqrt{5} - 1}{4})}^{2} = - \frac{1}{2^{14}}$