<s>[INST] Evaluate $\sec \frac{2\pi}{7}+\sec \frac{4\pi}{7}+\sec \frac{6\pi}{7}$ The problem is as follows:
Find the value of $\textrm{H}$ which belongs to a certain vibration coming from a magnet.
$$H=\sec \frac{2\pi}{7}+\sec \frac{4\pi}{7}+\sec \frac{6\pi}{7}$$
It was easy to spot that each term was related to multiples of two and three of the first angle. So I rewrote that equation like this:
$$H=\sec \frac{2\pi}{7}+\sec \frac{2\times 2\pi}{7}+\sec \frac{3\times 2\pi}{7}$$
One method which I tried was to transform the multiples of each angle into their equivalents as a single one as shown below:
$$\cos^{2}\omega=\frac{1+\cos 2\omega}{2}$$
$$\cos 2\omega= 2 \cos^{2}\omega - 1$$
$$\cos^{3}\omega=\frac{1}{4}\left(3cos\omega+\cos 3\omega \right)$$
$$\cos 3\omega = 4 \cos^{3}\omega - 3 cos\omega$$
Therefore by plugin these expressions into the above equation would become into (provided that secant function is expressed in terms of secant):
$$H=\frac{1}{\cos \frac{2\pi}{7}}+\frac{1}{2\cos^{2}\frac{2\pi}{7}-1}+\frac{1}{4\cos^{3}\frac{2\pi}{7}-3\cos\omega}$$
But from here on it looks convoluted or too algebraic to continue. My second guess was it could be related to sum to product identity but I couldn't find one for the secant.
Does it exist a shortcut or could it be that am I missing something? Can somebody help me to find the answer?
Can this problem be solved without requiring to use Euler's formulas?
 [/INST] If $z=e^{2\pi i/7}$, then
$$
\cos\frac{2n\pi}{7}=\frac{z^n+z^{-n}}{2}=\frac{z^{2n}+1}{2z^n}
$$
so your expression becomes
$$
\frac{2z}{z^2+1}+\frac{2z^2}{z^4+1}+\frac{2z^3}{z^6+1}
$$
We get the numerator
$$
2z(z^{10}+z^4+z^6+1+z^9+z^3+z^7+z+z^8+z^4+z^6+z^2)
$$
Now we can note that $z^7=1$ and $z^6+z^5+z^4+z^3+z^2+z+1=0$, so the expression becomes
$$
4z(z^6+z^4+z^3+z^2+z+1)=-4z^6
$$
The denominator is
\begin{align}
(z^2+1)(z^{10}+z^6+z^4+1)
&=z^{12}+z^8+z^6+z^2+z^{10}+z^6+z^4+1\\
&=z^5+z+z^6+z^2+z^3+z^6+z^4+1\\
&=z^6
\end{align}
</s>