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Question Number 197073 by Abdullahrussell last updated on 07/Sep/23

Answered by Frix last updated on 07/Sep/23

$$\sin \frac{\pi }{7}\sin \frac{2\pi }{7}\sin \frac{3\pi }{7}=x$$$$\mathrm{Use}\mathrm{trigonometric}\mathrm{formulas}\mathrm{to}\mathrm{get}$$$$\frac{1}{4}(\cos \frac{3\pi }{14}+\cos \frac{\pi }{14}-\sin \frac{\pi }{7})=x\left(1\right)$$$$\mathrm{Now}\mathrm{comes}\mathrm{the}‘‘{\mathrm{trick}}^{″}$$$${(\cos \frac{3\pi }{14}+\cos \frac{\pi }{14}-\sin \frac{\pi }{7})}^3=64x^3$$$$\mathrm{Again}\mathrm{use}\mathrm{trigonometric}\mathrm{formulas}\mathrm{to}\mathrm{get}$$$$\frac{7}{4}(\cos \frac{3\pi }{14}+\cos \frac{\pi }{14}-\sin \frac{\pi }{7})=64x^3\left(2\right)$$$$\Rightarrow$$$$4x=\frac{256x^3}{7}$$$$\mathrm{we}\mathrm{know}x>0\Rightarrow x=\frac{\sqrt{7}}{8}$$

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