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Question Number 189256 by BaliramKumar last updated on 13/Mar/23

$$Provethat$$$$\sin 10°=\frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-...........\mathrm{\infty }}}}}}}}}}}}}}$$

Answered by Frix last updated on 14/Mar/23

$$\sin 10°=\sin \frac{\pi }{18}$$$$\mathrm{First}{\mathrm{let}}^{\prime }\mathrm{s}\mathrm{find}x=\sqrt{2-\sqrt{2+\sqrt{2+x}}}$$$$3\mathrm{times}\mathrm{squaring}\mathrm{and}\mathrm{transforming}\mathrm{leads}\mathrm{to}$$$$x^8-8x^6+20x^4-16x^2-x+2=0$$$$\mathrm{Fortunately}x=-1\wedge x=2\mathrm{are}\mathrm{solutions}$$$$\left(\mathrm{but}{\mathrm{they}}^{\prime }\mathrm{re}\mathrm{both}\mathrm{false}\right)$$$$\mathrm{Now}{\mathrm{we}}^{\prime }\mathrm{ve}\mathrm{got}$$$$x^6+x^5-5x^4-3x^3+7x^2+x-1=0$$$$(x^3-3x+1)(x^3+x^2-2x-1)=0$$$$\mathrm{The}\mathrm{first}\mathrm{is}:$$$$(x+2\cos \frac{\pi }{9})(x-2\sin \frac{\pi }{18})(x-2\cos \frac{2\pi }{9})=0$$$$(\mathrm{The}\mathrm{second}\mathrm{is}\mathrm{irrelevant}.)$$$$\Rightarrow$$$$2\sin \frac{\pi }{18}=\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-...}}}}}}}$$

Commented by BaliramKumar last updated on 14/Mar/23

$$thankssir$$

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