First slide

Multiple and sub- multiple Angles

Question

The value of $\cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ , is

Moderate

A

$\frac{1}{2^{6}}$
B

$\frac{1}{2^{7}}$
C

$\frac{1}{2^{8}}$
D

none of these

Solution

We have,
$\begin{array}{l} \cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15} \\ = (\cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{7 π}{15}) \times (\cos \frac{3 π}{15} \cos \frac{6 π}{15}) \times \cos \frac{5 π}{15} \\ = \{\cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos (π - \frac{8 π}{15})\} (\cos \frac{3 π}{15} \cos \frac{6 π}{15}) \end{array}$
$\times \cos \frac{π}{3}$
$= (- \cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15}) \times (\cos \frac{3 π}{15} \cos \frac{6 π}{15}) \times \frac{1}{2}$
$= \frac{\sin (2^{4} \times \begin{matrix} π \\ 15 \end{matrix})}{2^{4} \sin \frac{π}{15}} \times \frac{\sin (2^{2} \times \begin{matrix} 3 π \\ 15 \end{matrix})}{2^{2} \sin \frac{3 π}{15}} \times \frac{1}{2}$
$= - \frac{\sin \frac{16 π}{15}}{16 \sin \frac{π}{15} \times \frac{\sin (\frac{12 π}{15})}{4 \sin \frac{3 π}{15}} \times \frac{1}{2} = \frac{1}{16} \times \frac{1}{4} \times \frac{1}{2} = \frac{1}{2^{7}}}$

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