Question-224562

Question Number 224562 by gregori last updated on 19/Sep/25

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Commented by Frix last updated on 19/Sep/25

r=(a_2 /a_1 )=(a_3 /a_2 ) ⇔ ((cos x)/(sin x))=((sin x)/(cos^2 x)) ⇔ ((cos^3 x)/(sin^2 x))=1 ⇔ determinant (((cos^3 x +cos^2 x −1=0))) a_1 =sin x a_2 =sin x ((cos x)/(sin x))=cos x a_3 =cos x ((cos x)/(sin x))=((cos^2 x)/(sin x)) a_4 =((cos^2 x)/(sin x))×((cos x)/(sin x))=((cos^3 x)/(sin^2 x))=1 ⇒ a_5 =(1/a_3 ) a_6 =(1/a_2 ) a_7 =(1/a_1 ) a_8 =(1/a_1 )×((cos x)/(sin x))=((cos x)/(sin^2 x))=((cos x)/(1−cos^2 x)) Testing ((cos x)/(1−cos^2 x))=1+cos x ⇔ cos x =(1−cos^2 x)(1+cos x) ⇔ determinant (((cos^3 x +cos^2 x −1=0))) ⇒ n=8

$${r}=\frac{{a}_{\mathrm{2}} }{{a}_{\mathrm{1}} }=\frac{{a}_{\mathrm{3}} }{{a}_{\mathrm{2}} }\:\Leftrightarrow\:\frac{\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}}=\frac{\mathrm{sin}\:{x}}{\mathrm{cos}^{\mathrm{2}} \:{x}}\:\Leftrightarrow\:\frac{\mathrm{cos}^{\mathrm{3}} \:{x}}{\mathrm{sin}^{\mathrm{2}} \:{x}}=\mathrm{1} \\ $$$$\Leftrightarrow\:\begin{array}{|c|}{\mathrm{cos}^{\mathrm{3}} \:{x}\:+\mathrm{cos}^{\mathrm{2}} \:{x}\:−\mathrm{1}=\mathrm{0}}\\\hline\end{array} \\ $$$$ \\ $$$${a}_{\mathrm{1}} =\mathrm{sin}\:{x} \\ $$$${a}_{\mathrm{2}} =\mathrm{sin}\:{x}\:\frac{\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}}=\mathrm{cos}\:{x} \\ $$$${a}_{\mathrm{3}} =\mathrm{cos}\:{x}\:\frac{\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}}=\frac{\mathrm{cos}^{\mathrm{2}} \:{x}}{\mathrm{sin}\:{x}} \\ $$$${a}_{\mathrm{4}} =\frac{\mathrm{cos}^{\mathrm{2}} \:{x}}{\mathrm{sin}\:{x}}×\frac{\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}}=\frac{\mathrm{cos}^{\mathrm{3}} \:{x}}{\mathrm{sin}^{\mathrm{2}} \:{x}}=\mathrm{1} \\ $$$$\Rightarrow \\ $$$${a}_{\mathrm{5}} =\frac{\mathrm{1}}{{a}_{\mathrm{3}} } \\ $$$${a}_{\mathrm{6}} =\frac{\mathrm{1}}{{a}_{\mathrm{2}} } \\ $$$${a}_{\mathrm{7}} =\frac{\mathrm{1}}{{a}_{\mathrm{1}} } \\ $$$${a}_{\mathrm{8}} =\frac{\mathrm{1}}{{a}_{\mathrm{1}} }×\frac{\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}}=\frac{\mathrm{cos}\:{x}}{\mathrm{sin}^{\mathrm{2}} \:{x}}=\frac{\mathrm{cos}\:{x}}{\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \:{x}} \\ $$$$\mathrm{Testing}\:\frac{\mathrm{cos}\:{x}}{\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \:{x}}=\mathrm{1}+\mathrm{cos}\:{x} \\ $$$$\Leftrightarrow \\ $$$$\mathrm{cos}\:{x}\:=\left(\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \:{x}\right)\left(\mathrm{1}+\mathrm{cos}\:{x}\right)\:\Leftrightarrow\:\begin{array}{|c|}{\mathrm{cos}^{\mathrm{3}} \:{x}\:+\mathrm{cos}^{\mathrm{2}} \:{x}\:−\mathrm{1}=\mathrm{0}}\\\hline\end{array} \\ $$$$\Rightarrow\:{n}=\mathrm{8} \\ $$

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