show that one fo the roots of the equation $\displaystyle x^3-6x^2+9x-1=0$ is $\displaystyle 4\sin^2\frac{\pi}{18}$ and find the other two roots
Let $\displaystyle x=y+2$.
This gives, $\displaystyle (y+2)^3 - 6(y+2)^2 + 9(y+2) - 1 = 0$.
Which simplifies to $\displaystyle y^3-3y+1=0$.
Let $\displaystyle C = - \frac{\Delta}{108} = - \frac{3}{4}$
If $\displaystyle A = -\frac{1}{2} + \sqrt{C} = e^{2\pi i/3}$ and $\displaystyle B = -\frac{1}{2} - \sqrt{C} = e^{-2\pi i/3}$.
Then this means $\displaystyle u=\sqrt[3]{A} = e^{2\pi i/9}$ and $\displaystyle v=\sqrt[3]{B} = e^{-2\pi i/9}$ would produce solutions:
$\displaystyle u+v,u\zeta + v\zeta^2,u\zeta^2+v\zeta$ (where $\displaystyle \zeta = e^{2\pi i/3})$
Simplification of these formulas leads to the solutions:
$\displaystyle 2\cos \frac{2\pi}{9}, 2\cos \frac{8\pi}{9}, 2\cos \frac{14\pi}{9}$.
And so the three solutions to the original are:
$\displaystyle x_1 = 2 + 2\cos \frac{2\pi}{9} = 4\sin^2 \frac{\pi}{18}$
$\displaystyle x_2 = 2 + 2\cos \frac{8\pi}{9} = 4\sin^2 \frac{4\pi}{18}$
$\displaystyle x_3 = 2 + 2\cos \frac{14\pi}{9}= 4\sin^2 \frac{7\pi}{18}$
Use the formulas $\displaystyle \cos(2\theta) = 1 - 2\sin^2\theta$ and $\displaystyle \cos(3\theta) = 4\cos^3\theta - 3\cos\theta$. These show that if $\displaystyle x=4\sin^2\theta$ then $\displaystyle \cos(2\theta) = 1 - \tfrac x2$, and so $\displaystyle \cos(6\theta) = 4\left(1 - \tfrac x2\right)^3 - 3\left(1 - \tfrac x2\right) = 1 - \tfrac92x + 3x^2 - \tfrac12x^3$. But if $\displaystyle \theta = \tfrac\pi{18}$ then $\displaystyle \cos(6\theta)=\tfrac12$. That gives the equation $\displaystyle x^3-6x^2+9x-1=0$. The other two solutions will come from the other angles satisfying $\displaystyle \cos(6\theta)=\tfrac12$, namely $\displaystyle \theta = \tfrac{5\pi}{18}$ and $\displaystyle \theta = \tfrac{7\pi}{18}$. (ThePerfectHacker had $\displaystyle \theta = \tfrac{4\pi}{18}$, but I think $\displaystyle \theta = \tfrac{5\pi}{18}$ must be correct.)
Yes I made a mistake. But we are going to keep it secret.
The mistake was in the simplification. The solutions that I got to the depressed cubic are correct I just made mistakes in the identities.
Here is the corrected version:
1)$\displaystyle 2 + 2\cos \frac{2\pi}{9} = 2 - 2\cos \frac{7\pi}{9} = 4\sin^2 \frac{7\pi}{18}$
2)$\displaystyle 2 + 2\cos \frac{8\pi}{9} = 2 - 2\cos \frac{\pi}{9} = 4\sin^2 \frac{\pi}{18}$
3)$\displaystyle 2 + 2\cos \frac{14\pi}{9} = 2 - 2\cos \frac{5\pi}{9} = 4\sin^2 \frac{5\pi}{18}$