riwayatcbt

2022-11-25

Given:
$x+\frac{1}{x}=2\mathrm{cos}\left(\mathrm{cos}\theta \right)$
Prove:
${x}^{n}+\frac{1}{{x}^{n}}=2\mathrm{cos}\left(\mathrm{cos}\left(n\theta \right)\right)$

Bria Mccoy

Expert

When $x\in \mathbb{R}\setminus 0$ we have that $x+\frac{1}{x}\in \right]-\mathrm{\infty },-2\right]\cup \left[2,+\mathrm{\infty }\left[$. On the other hand, $2\mathrm{cos}\left(\mathrm{cos}\theta \right)\in \left[2\mathrm{cos}1,2\right]$. So, the only way that you can have $x+\frac{1}{x}=2\mathrm{cos}\left(\mathrm{cos}\theta \right)$ is with $x=1$ and $\mathrm{cos}\theta =0$
However, even for $x=1$ and $\mathrm{cos}\theta =0$, the statement does not hold for every n.

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