Rational Solutions to Trigonometric Equation

Question

Consider the equation $$ \cos(\pi a) + \cos(\pi b) = \cos(\pi c) + \cos(\pi d), $$ with $$ a,b,c,d \in \mathbb{Q} \cap \left[0,\frac{1}{2}\right]. $$ Clearly, this equation admits some trivial solutions, namely $(a,b) = (c,d)$ or $(a,b) = (d,c)$. Are there any rational solutions other than the trivial ones?

Answer 1

For nontrivial solutions, $a,b,c,d$ are distinct. WLOG let $a = \max(a,b,c,d)$.

Taking a common denominator $N$ and writing $a=A/N$, ..., $d = D/N$, we can write this as $$ \omega^A + \omega^{-A} + \omega^B + \omega^{-B} - \omega^C - \omega^{-C} - \omega^D - \omega^{-D} = 0$$ or equivalently $$ P(\omega) = \omega^{2A} + 1 + \omega^{B+A} + \omega^{-B+A} - \omega^{C+A} - \omega^{-C+A} - \omega^{D+A} - \omega^{-D+A} = 0 $$ where $\omega = \exp(\pi i/N)$. Now the minimal polynomial of $\omega$ over the rationals is the cyclotomic polynomial $C_{2N}(z)$, so $P(z)$ must be divisible by $C_{2N}(z)$. I searched over $A \le 20$, in each case factoring $P(z)$ and looking for cyclotomic factors $C_M(z)$ with $M \ge 4A$, obtaining the following results.

$$ \eqalign{ \cos(\pi/3) + \cos(\pi/15) &= \cos(4 \pi/15) + \cos(\pi/5)\cr \cos(\pi/2) + \cos(\pi/12) &= \cos(5 \pi/12) + \cos(\pi/4)\cr \cos(\pi/2) + \cos(\pi/18) &= \cos(7 \pi/18) + \cos(5 \pi/18)\cr \cos(\pi/2) + \cos(\pi/9) &= \cos(4 \pi/9) + \cos(2 \pi/9)\cr \cos(3 \pi/7) + \cos(\pi/7) &= \cos(\pi/3) + \cos(2 \pi/7)\cr \cos(\pi/2) + \cos(\pi/24) &= \cos(3 \pi/8) + \cos(7 \pi/24)\cr \cos(\pi/2) + \cos(\pi/8) &= \cos(11 \pi/24) + \cos(5 \pi/24)\cr \cos(\pi/2) + \cos(\pi/30) &= \cos(11 \pi/30) + \cos(3 \pi/10)\cr \cos(\pi/2) + \cos(\pi/15) &= \cos(2 \pi/5) + \cos(4 \pi/15)\cr \cos(\pi/2) + \cos(\pi/10) &= \cos(13 \pi/30) + \cos(7 \pi/30)\cr \cos(\pi/2) + \cos(2 \pi/15) &= \cos(7 \pi/15) + \cos(\pi/5)\cr \cos(\pi/2) + \cos(\pi/5) &= \cos(2 \pi/5) + \cos(\pi/3)\cr \cos(\pi/2) + \cos(\pi/36) &= \cos(13 \pi/36) + \cos(11 \pi/36)\cr \cos(\pi/2) + \cos(5 \pi/36) &= \cos(17 \pi/36) + \cos(7 \pi/36)\cr } $$

Those involving $\cos(\pi/2)$ might be considered as somewhat trivial: they are cases of $$\cos(t) = \cos\left(\frac{\pi}{3} + t\right) + \cos(\left(\frac{\pi}{3}-t\right)$$

So the "really nontrivial" examples are $$\eqalign{\cos(\pi/3) + \cos(\pi/15) &= \cos(4 \pi/15) + \cos(\pi/5)\cr \cos(3 \pi/7) + \cos(\pi/7) &= \cos(\pi/3) + \cos(2 \pi/7)\cr }$$

Are there more?

Answer 2

Yes :

Set $\ a=\frac{1}{3}$ , $b=\frac{2}{5}$ , $c=\frac{1}{5}$ , $d=\frac{1}{2}$