Question

Write the value of $\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \cos \frac{\mathrm{\pi }}{7}\cos \frac{2\mathrm{\pi }}{7}\cos \frac{4\mathrm{\pi }}{7}=\frac{2\sin {\displaystyle \frac{\mathrm{\pi }}{7}}\cos {\displaystyle \frac{\mathrm{\pi }}{7}}\cos {\displaystyle \frac{2\mathrm{\pi }}{7}}\cos {\displaystyle \frac{4\mathrm{\pi }}{7}}}{2\sin {\displaystyle \frac{\mathrm{\pi }}{7}}} \\ \left[\mathrm{On}\mathrm{dividing}\mathrm{and}\mathrm{multiplying}\mathrm{by}2\sin \frac{\mathrm{\pi }}{7}\right] \\ =\frac{2\times \sin {\displaystyle \frac{2\mathrm{\pi }}{7}}\cos {\displaystyle \frac{2\mathrm{\pi }}{7}}\cos {\displaystyle \frac{4\mathrm{\pi }}{7}}}{2\times 2\sin {\displaystyle \frac{\mathrm{\pi }}{7}}} \end{gathered}$$

Proceeding in the same way, we get

$$\begin{gathered} \cos \frac{\mathrm{\pi }}{7}\cos \frac{2\mathrm{\pi }}{7}\cos \frac{4\mathrm{\pi }}{7}=\frac{\sin {\displaystyle \frac{8\mathrm{\pi }}{7}}}{8\sin {\displaystyle \frac{\mathrm{\pi }}{7}}} \\ =\frac{\sin {\displaystyle \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{7}\right)}}{8\sin {\displaystyle \frac{\mathrm{\pi }}{7}}} \\ =\frac{-\sin {\displaystyle \frac{\mathrm{\pi }}{7}}}{8\sin \frac{\mathrm{\pi }}{7}} \\ \therefore \cos \frac{\mathrm{\pi }}{7}\cos \frac{2\mathrm{\pi }}{7}\cos \frac{4\mathrm{\pi }}{7}=\frac{-1}{8} \end{gathered}$$