Prove $\frac{1}{4-\sec^{2}\frac{2\pi}{7}} + \frac{1}{4-\sec^{2}\frac{4\pi}{7}} + \frac{1}{4-\sec^{2}\frac{6\pi}{7}} = 1.$

Question

How can I prove the fact $$\frac{1}{4-\sec^{2}\frac{2\pi}{7}} + \frac{1}{4-\sec^{2}\frac{4\pi}{7}} + \frac{1}{4-\sec^{2}\frac{6\pi}{7}} = 1.$$

When asked somebody told me to use the ideas of Chebyshev polynomial, but I haven't learnt that in school.

I tried doing this way:

Look at $y =\cos\theta + i \sin\theta$ where $\displaystyle\theta \in \Bigl\{\frac{2\pi}{7},\frac{4\pi}{7},\cdots,2\pi\Bigr\}$

Then we have \begin{align*} y^{7} &=1 \\ y^{7}-1 &=0 \\ (y-1) \cdot (y^{6}+y^{5}+\cdots + 1) &= 0 \end{align*}

Now the root $y=1$ corresponds to $\theta = 2\pi$, and that $$y^{6} + y^{5}+\cdots + 1 =0$$

have roots $\cos\theta + i \sin\theta$, where $\theta \in \Bigl\{\frac{2\pi}{7},\frac{4\pi}{7} ,\cdots \Bigr\}$. Looking at $y+\frac{1}{y} $ will give me the roots as $\cos\theta$ and then i can put $z=y^{2}$ to get $\cos^{2}$ as the roots and the invert to get $\sec^{2}$, but I have some problems.

Can anyone help me out with a neat solution. Thanks.

Answer 1

Note that, for every $x$ such that this fraction exists, $$ \frac1{4-\sec^2x}=\frac14+\frac14\frac1{2\cos(2x)+1}=\frac14+\frac14\frac{\mathrm e^{2\mathrm ix}}{\mathrm e^{4\mathrm ix}+\mathrm e^{2\mathrm ix}+1}, $$ hence the identity to be proved is $$ \sum\limits_z\frac{z^2}{z^4+z^{2}+1}=1, $$ where the sum is over $z$ in $\{\omega,\omega^2,\omega^3\}$ with $\omega=\mathrm e^{2\mathrm i\pi/7}$.

If $z^7=1$ and $z\ne1$, then $\dfrac{z^2}{z^4+z^2+1}=-z^3-z^{4}$ hence the sum in the LHS is $$ -\omega^3-\omega^{4}-\omega^6-\omega^{8}-\omega^9-\omega^{12}, $$ that is, $$ -\omega^3-\omega^{4}-\omega^6-\omega-\omega^2-\omega^5=1-\sum\limits_{k=1}^7\omega^k=1. $$

Answer 2

Where you have left of $y^6+y^5+\cdots+y+1=0$ where $y=\cos \theta+i\sin \theta$ where $\theta=\frac{2\pi}7,\frac{4\pi}7,\frac{6\pi}7,\cdots \frac{12\pi}7$

Let us divide both sides by $y^3,$ $y^3+\frac1{y^3}+y^2+\frac 1{y^2}+\frac 1 y+1=0$

or $(y+\frac 1 y)^3-3(y+\frac 1 y)+(y+\frac 1 y)^2-2+(y+\frac 1 y)+1=0$

or $(y+\frac 1 y)^3+(y+\frac 1 y)^2-3(y+\frac 1 y)-1=0$

Now, $y+\frac 1 y=2\cos \theta=z$(say)

So, $z^3+z^2-3z-1=0--->(1), z^2(1+z)=3z-1,z^2=\frac{3z-1}{z+1}$

Now, $$\frac 1{4-\sec^2\theta}=\frac{\cos^2\theta}{4\cos^2\theta-1}= \frac{z^2}{4z^2-4}=w(say),$$

So, $z^2=\frac {4w}{4w-1}$

Comparing the values of $z^2, \frac {4w}{4w-1}=\frac{3z-1}{z+1}$

Replacing the $z$ with $w$ in (1), we shall get a cubic equation in $w,$ whose sum of roots will give us the required identity.