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Question Number 108222 by ZiYangLee last updated on 15/Aug/20

Find the value of tan6°tan42°tan66°tan78° without calculator.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\mathrm{tan6}°\mathrm{tan42}°\mathrm{tan66}°\mathrm{tan78}°\:\mathrm{without} \\ $$$$\mathrm{calculator}. \\ $$

Answered by Dwaipayan Shikari last updated on 15/Aug/20

sin6°sin42°sin66°sin78° =(1/4)(cos60°−cos72°)(cos36°−cos120°) =(1/4)((1/2)−(((√5)−1)/4))((((√5)+1)/4)+(1/2)) =(1/4)(((3−(√5))/4))((((√5)+3)/4))=(1/(16)) cos6°cos42°cos66°cos78° =(1/4)(cos60°+cos72°)(cos120°+cos36°) =(1/4)((1/2)+(((√5)−1)/4))(−(1/2)+(((√5)+1)/4)) =(1/4)((((√5)+1)/4))((((√5)−1)/4))=(1/(16)) so tan6°tan42°tan66°tan78°=1

$${sin}\mathrm{6}°{sin}\mathrm{42}°{sin}\mathrm{66}°{sin}\mathrm{78}° \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left({cos}\mathrm{60}°−{cos}\mathrm{72}°\right)\left({cos}\mathrm{36}°−{cos}\mathrm{120}°\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}\right)\left(\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{3}−\sqrt{\mathrm{5}}}{\mathrm{4}}\right)\left(\frac{\sqrt{\mathrm{5}}+\mathrm{3}}{\mathrm{4}}\right)=\frac{\mathrm{1}}{\mathrm{16}} \\ $$$${cos}\mathrm{6}°{cos}\mathrm{42}°{cos}\mathrm{66}°{cos}\mathrm{78}° \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left({cos}\mathrm{60}°+{cos}\mathrm{72}°\right)\left({cos}\mathrm{120}°+{cos}\mathrm{36}°\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}\right)\left(−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}\right)\left(\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}\right)=\frac{\mathrm{1}}{\mathrm{16}} \\ $$$${so} \\ $$$${tan}\mathrm{6}°{tan}\mathrm{42}°{tan}\mathrm{66}°{tan}\mathrm{78}°=\mathrm{1} \\ $$

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