If x + (1)/(x) = 2 θ , then what is x^3 + (1)/(x^3) equal to?

Given $x + \dfrac{1}{x} = 2\cos\theta$

Using the algebraic identity $a^3 + b^3 = (a+b)^3 - 3ab(a+b)$:

x^3 + \dfrac{1}{x^3} = \left(x + \dfrac{1}{x}\right)^3 - 3 \left(x + \dfrac{1}{x}\right)<br><br>Substituting x + $\dfrac{1}{x} =2\$ cos\theta:<br><br>$ x^3 + \dfrac{1}{x^3} = ( 2\cos\theta)^3 -3(2\cos\theta)$<br><br>$x^3 + \dfrac{1}{x^3} =8\$\cos^3\theta -$6\$cos\theta<br><br>$x^3 + \dfrac{1}{x^3} = 2(4\cos^3\theta -3\cos\theta)$<br><br>Using the trigonometric identity$\cos 3\theta =4\$\cos^3\theta -$3\$cos\theta:<br><br>$x^3 + \dfrac{1}{x^3} =$2$\cos 3\theta$<br><br>Answer: 2$\cos 3\theta