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Question Number 168277 by Eulerian last updated on 07/Apr/22

$$\mathrm{Prove}\mathrm{that}\frac{\sin \left(\frac{3\pi }{5}\right)}{\sin \left(\frac{4\pi }{5}\right)}=\frac{1+\sqrt{5}}{2}$$

Commented by cortano1 last updated on 07/Apr/22

$$\frac{\sin 108°}{\sin 144°}=\frac{\sin (90°+18°)}{\sin (180°-36°)}$$$$=\frac{\cos 18°}{\sin 36°}=\frac{\cos 18°}{2\sin 18°\cos 18°}$$$$=\frac{1}{2\sin 18°}=\frac{1}{2\left(\frac{\sqrt{5}-1}{4}\right)}$$$$=\frac{2}{\sqrt{5}-1}\times \frac{\sqrt{5}+1}{\sqrt{5}+1}=\frac{\sqrt{5}+1}{2}$$

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