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I'm using LatticeReduce to find linear relationships in these trigonometric expressions,

vec = Table[1/Inactive[Sin][k Pi/15], {k, 7}];
mat = LatticeReduce[Join[IdentityMatrix[Length@#], List /@ -Round[10^50 #], 2]] &@
   Activate@vec;

Thread[Select[mat, Norm@# < 10 &][[All, 1 ;; -2]] . vec == 0] // Simplify // Column

$\begin{array}{l} \frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{2}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{2}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{1}{\sin \left(\frac{\pi }{5}\right)}=\frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)} \\ \end{array}$

It seems that the result is not bad. However, there are still some relationships that have not been found. For example:

$\begin{array}{l} \frac{1}{\sin \left(\frac{7 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{15}\right)}=\frac{3}{\sin \left(\frac{\pi }{3}\right)} \\ \frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{2}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{2}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{\pi }{5}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)} \\ \end{array}$

Is there any way to find more relationships? Maybe we can do a full permutation of the vec, but for larger vec, it's very slow.

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1 Answer 1

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If we limit vec to four elements and test all possible sublist we get more results.

vec = Subsets[Table[1/Inactive[Sin][k Pi/15], {k, 7}], {4}];

mat = LatticeReduce[
     Join[IdentityMatrix[Length@#], List /@ -Round[10^50 #], 2]] & /@ 
   Activate@vec;

Activate /@ 
  HoldForm /@ (Thread[
       Flatten[Map[f |-> #[[2]] . f, #[[1]]] & /@ 
          Transpose[{Select[#, Norm@# < 10 &][[All, 1 ;; -2]] & /@ 
             mat, vec}]] == 0] // Simplify // 
     DeleteDuplicates) // Column

$\begin{array}{l} \frac{2}{\sin \left(\frac{\pi }{15}\right)}+\frac{1}{\sin \left(\frac{4 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{5}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{5}{\sin \left(\frac{\pi }{5}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)}=\frac{3}{\sin \left(\frac{\pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{2}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{2}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{5}{\sin \left(\frac{\pi }{5}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{3}{\sin \left(\frac{\pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{5}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{3}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{2}{\sin \left(\frac{4 \pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)} \\ \frac{6}{\sin \left(\frac{\pi }{3}\right)}+\frac{5}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{3}{\sin \left(\frac{2 \pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{3}{\sin \left(\frac{\pi }{3}\right)} \\ \frac{5}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{2}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{15}\right)} \\ \frac{3}{\sin \left(\frac{\pi }{5}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{1}{\sin \left(\frac{4 \pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{2}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{5}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{3}{\sin \left(\frac{\pi }{5}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{2}{\sin \left(\frac{\pi }{15}\right)} \\ \frac{3}{\sin \left(\frac{\pi }{15}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{5}{\sin \left(\frac{\pi }{5}\right)}+\frac{6}{\sin \left(\frac{\pi }{3}\right)} \\ \frac{1}{\sin \left(\frac{\pi }{5}\right)}+\frac{2}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{5}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{3}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{2}{\sin \left(\frac{\pi }{15}\right)} \\ \frac{5}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{3}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)} \\ \frac{3}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{6}{\sin \left(\frac{\pi }{3}\right)}=\frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{5}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{1}{\sin \left(\frac{\pi }{5}\right)}+\frac{2}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{4 \pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{3}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{2}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{5}{\sin \left(\frac{\pi }{5}\right)} \\ 3 \left(\frac{1}{\sin \left(\frac{\pi }{3}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}\right)=\frac{2}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{3}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{3}{\sin \left(\frac{\pi }{3}\right)}+\frac{5}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{3}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{4 \pi }{15}\right)} \\ \frac{5}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{2}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{4 \pi }{15}\right)} \\ \frac{3}{\sin \left(\frac{\pi }{5}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{2}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)} \\ \frac{3}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{5}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{2}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{3}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{2}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{\pi }{5}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)} \\ \frac{5}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{3}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{6}{\sin \left(\frac{\pi }{3}\right)} \\ \end{array}$

With five elements in sub-lists of vec we get additional identities.

$\begin{array}{l} \frac{1}{\sin \left(\frac{\pi }{5}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)}+\frac{2}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{15}\right)} \\ \frac{1}{\sin \left(\frac{\pi }{15}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{2}{\sin \left(\frac{\pi }{5}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)} \\ \frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)}+\frac{1}{\sin \left(\frac{2 \pi }{5}\right)}=\frac{1}{\sin \left(\frac{2 \pi }{15}\right)}+\frac{2}{\sin \left(\frac{\pi }{5}\right)} \\ \frac{1}{\sin \left(\frac{\pi }{5}\right)}+\frac{2}{\sin \left(\frac{2 \pi }{5}\right)}+\frac{1}{\sin \left(\frac{7 \pi }{15}\right)}=\frac{1}{\sin \left(\frac{4 \pi }{15}\right)}+\frac{3}{\sin \left(\frac{\pi }{3}\right)} \\ \end{array}$

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