Prove that solution of the equation $8x^3-4x^2-4x+1= 0$ has roots $\cos\frac{\pi}{7},\cos\frac{3 \pi}{7},\cos\frac{5 \pi}{7}$

Question

Prove that solution of the equation $8x^3-4x^2-4x+1= 0$ has roots $\cos\frac{\pi}{7},\cos\frac{3 \pi}{7}\space \text{and}\space \cos\frac{5 \pi}{7}$.

How to even solve it ? I have no idea.

Because cubic have no formula like quadratic equations to obtain the roots and the expression like $\cos\frac{\pi}{7}$ is something that I don't think I can evaluate with my limited knowledge (I'm in highschool) of only $\cos 2x$ or $\cos 3x$ type. Since it has denominator of 7, I thought maybe the seventh root of unity might help me, but since only some of its multiples are used, I don't think so it will be of much use.

How do I solve this problem. Please help.

Answer 1

You could prove it with just a couple of high-school trigonometric identities.

Let $\theta = \pi/7 $ and recognize $$x_1=\cos \theta = -\cos 6\theta$$ $$x_2=\cos3\theta = -\cos 4\theta$$ $$x_3=\cos5\theta = -\cos 2\theta$$

(1). Evaluate their product by applying the identity $\sin 2x = 2\sin x \cos x$ $$ \cos2\theta \cos4\theta \cos6\theta = \frac{\sin 4\theta \cos 4\theta\cos 6\theta}{2\sin 2\theta} = \frac{\sin 8\theta \cos 8\theta }{4\sin 2\theta}= \frac{1}{8}$$

(2). Evaluate their sum by applying $\sin(x+y)+\sin(x-y)=2\sin x\cos y$,

$$ 2\sin 2\theta(\cos2\theta + \cos4\theta + \cos6\theta)$$ $$=\sin 4\theta + (\sin 6\theta - \sin 2\theta) + (\sin 8\theta - \sin 4\theta)$$

which after some cancellation leads to, $$\cos2\theta + \cos4\theta + \cos6\theta = -\frac{1}{2}$$

(3). Evaluate the sum of their cross products by applying $\cos(x+y)+\cos(x-y)=2\cos x\cos y$,

$$\cos4\theta \cos6\theta + \cos6\theta \cos2\theta + \cos2\theta \cos4\theta $$ $$= \cos2\theta + \cos4\theta + \cos6\theta = -\frac{1}{2}$$

So, in terms of $x_1$, $x_2$ and $x_3$,

$$x_1+x_2+x_3=\frac{1}{2}$$ $$x_1x_2+x_2x_3+x_3x_1=-\frac{1}{2}$$ $$x_1x_2x_3=-\frac{1}{8}$$

And

$$(x-x_1)(x-x_2)(x-x_3)=x^3 - \frac{1}{2}x^2-\frac{1}{2}x+\frac{1}{8}$$

Therefore, $x_1=\cos \theta$, $x_2=\cos 3\theta$ and $ x_3=\cos 5\theta$ are the roots of

$$x^3-\frac{1}{2}x^2-\frac{1}{2}x+\frac{1}{8}=0$$

or,

$$8x^3-4x^2-4x+1= 0$$

Answer 2

$$\cos\frac{\pi}{7}\cos\frac{3\pi}{7}\cos\frac{5\pi}{7}=\frac{8\sin\frac{\pi}{7}\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}}{8\sin\frac{\pi}{7}}=\frac{\sin\frac{8\pi}{7}}{8\sin\frac{\pi}{7}}=-\frac{1}{8}.$$ The rest is a similar and use the Viete's theorem.