Question

$x+\frac{1}{x}=2\cos \theta$ , then

$x^3+\frac{1}{x^3}$=?

Answer 1

$2\cos \theta =x+\frac{1}{x}$

$(x^3+\frac{1}{x^3})=?\Rightarrow \cos 3\theta =4\cos 3\theta -3\cos \theta$

$2\cos 3\theta =8\cos 3\theta -6\cos \theta$

$2\cos \theta =x+\frac{1}{x}$

$2\cos 3\theta ={(k+\frac{1}{x})}^3-3(1+\frac{1}{x})$

$2\cos 3\theta =x^3+3x^2\left(\frac{1}{x}\right)+3x\left(\frac{1}{x^2}\right)+\frac{1}{x^3}-3(x+\frac{1}{x})$

$=x^3+3x+\frac{3}{x}+\frac{1}{x^3}-3x-\frac{3}{x}$

$\cos 3\theta =x^3+\frac{1}{x^3}$