Q61 / 160 · AP EAMCET 2024 (22 May Shift 1) · Mathematics

$4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2 \pi}{7} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{7}=$

  • Correct answer
    A
    $-\frac{1}{8}$
  • B
    $\frac{1}{32}$
  • C
    $-\frac{1}{32}$
  • D
    $\frac{1}{8}$

Solution

$\begin{aligned} & 4 \cos \left(\frac{\pi}{7}\right) \cos \left(\frac{2 \pi}{7}\right) \cos \left(\frac{4 \pi}{7}\right) \times \cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \\ & \begin{array}{c}=\frac{2}{\sin \left(\frac{\pi}{7}\right)} \sin \left(\frac{2 \pi}{7}\right) \cos \left(\frac{2 \pi}{7}\right) \cos \left(\frac{4 \pi}{7}\right) \\ \times \frac{1}{2 \sin \left(\frac{\pi}{5}\right)} \sin \left(\frac{2 \pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right) \\ =\frac{1 \sin \left(\frac{8 \pi}{7}\right)}{2 \sin \left(\frac{\pi}{7}\right)} \times \frac{1}{4 \sin \left(\frac{\pi}{5}\right)} \sin \left(\frac{4 \pi}{5}\right)=-\frac{1}{8}\end{array}\end{aligned}$

Hence, the correct answer is option: 'A'.

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