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Question Number 165428 by mathlove last updated on 01/Feb/22

Answered by som(math1967) last updated on 01/Feb/22

$$\frac{sin6sin42sin66sin78}{cos6cos42cos66cos78}★$$$$=\frac{\left(2sin6sin66\right)\left(2sin42sin78\right)}{\left(2cos6cos66\right)\left(2cos42cos78\right)}$$$$=\frac{\{cos(66-6)-cos(66+6)\}\{cos36-cos120\}}{(cos72+cos60)(cos120+cos36)}$$$$=\frac{(\frac{1}{2}-\frac{\sqrt{5}-1}{4})(\frac{\sqrt{5}+1}{4}+\frac{1}{2})}{(\frac{\sqrt{5}-1}{4}+\frac{1}{2})(\frac{\sqrt{5}+1}{4}-\frac{1}{2})}★★$$$$=\frac{(3-\sqrt{5})(3+\sqrt{5})}{(\sqrt{5}+1)(\sqrt{5}-1)}=\frac{9-5}{5-1}=1$$$$★\frac{\mathit{\pi }}{30}=6°$$$$★★\mathit{c}\mathit{o}\mathit{s}72=\frac{\sqrt{5}-1}{4}\mathit{c}\mathit{o}\mathit{s}72=\frac{\sqrt{5}+1}{4}$$

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