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Question Number 189463 by Rupesh123 last updated on 17/Mar/23

Answered by witcher3 last updated on 17/Mar/23

cos(60+20) cos(40)=cos(60−20) =(1/2)(cos(20)+(√3)sin(20)) cos80=(1/2)(cos(20)−(√3)sin(20)) =((3cos(20)−1)^2 −27sin^2 (20))(6cos(20)+1) =(36cos^2 (20)−6cos(20)−26)(6cos(20)+1) c=cos20⇔ =216c^3 −162c−26=1 216c^3 −162c−27=0 ⇔8c^3 −6c−1=0...(E) cos(3.20)=4cos^3 (20)−3cos(20)=(1/2) ⇔8cos^3 (20)−6cos(20)−1=0 heins E True⇔ (6cos80−1)(6cos(40)−1)(6cos(20)+1)=1 c

$$\mathrm{cos}\left(\mathrm{60}+\mathrm{20}\right) \\ $$$$\mathrm{cos}\left(\mathrm{40}\right)=\mathrm{cos}\left(\mathrm{60}−\mathrm{20}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\left(\mathrm{20}\right)+\sqrt{\mathrm{3}}\mathrm{sin}\left(\mathrm{20}\right)\right) \\ $$$$\mathrm{cos80}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\left(\mathrm{20}\right)−\sqrt{\mathrm{3}}\mathrm{sin}\left(\mathrm{20}\right)\right) \\ $$$$=\left(\left(\mathrm{3cos}\left(\mathrm{20}\right)−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{27sin}^{\mathrm{2}} \left(\mathrm{20}\right)\right)\left(\mathrm{6cos}\left(\mathrm{20}\right)+\mathrm{1}\right) \\ $$$$=\left(\mathrm{36cos}^{\mathrm{2}} \left(\mathrm{20}\right)−\mathrm{6cos}\left(\mathrm{20}\right)−\mathrm{26}\right)\left(\mathrm{6cos}\left(\mathrm{20}\right)+\mathrm{1}\right) \\ $$$$\mathrm{c}=\mathrm{cos20}\Leftrightarrow \\ $$$$=\mathrm{216c}^{\mathrm{3}} −\mathrm{162c}−\mathrm{26}=\mathrm{1} \\ $$$$\mathrm{216c}^{\mathrm{3}} −\mathrm{162c}−\mathrm{27}=\mathrm{0} \\ $$$$\Leftrightarrow\mathrm{8c}^{\mathrm{3}} −\mathrm{6c}−\mathrm{1}=\mathrm{0}...\left(\mathrm{E}\right) \\ $$$$\mathrm{cos}\left(\mathrm{3}.\mathrm{20}\right)=\mathrm{4cos}^{\mathrm{3}} \left(\mathrm{20}\right)−\mathrm{3cos}\left(\mathrm{20}\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Leftrightarrow\mathrm{8cos}^{\mathrm{3}} \left(\mathrm{20}\right)−\mathrm{6cos}\left(\mathrm{20}\right)−\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{heins}\:\mathrm{E}\:\:\:\mathrm{True}\Leftrightarrow \\ $$$$\left(\mathrm{6cos80}−\mathrm{1}\right)\left(\mathrm{6cos}\left(\mathrm{40}\right)−\mathrm{1}\right)\left(\mathrm{6cos}\left(\mathrm{20}\right)+\mathrm{1}\right)=\mathrm{1} \\ $$$$\mathrm{c} \\ $$

Commented by Rupesh123 last updated on 18/Mar/23

Excellent!

Commented by witcher3 last updated on 18/Mar/23

thank You God bless You

$$\mathrm{thank}\:\mathrm{You}\:\mathrm{God}\:\mathrm{bless}\:\mathrm{You} \\ $$

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