{"paper":{"title":"Trigonometric identities and quadratic residues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.CA","authors_text":"Zhi-Wei Sun","submitted_at":"2019-08-06T15:59:18Z","abstract_excerpt":"In this paper we obtain some novel identities involving trigonometric functions. Let $n$ be any positive odd integer. We show that $$\\sum_{r=0}^{n-1}\\frac1{1+\\sin2\\pi\\frac{x+r}n+\\cos2\\pi\\frac{x+r}n} =\\frac{(-1)^{(n-1)/2}n}{1+(-1)^{(n-1)/2}\\sin 2\\pi x+\\cos 2\\pi x}$$ for any complex number with $x+1/2,x+(-1)^{(n-1)/2}/4\\not\\in\\mathbb Z$, and $$\\sum_{j,k=0}^{n-1}\\frac1{\\sin 2\\pi\\frac{x+j}n+\\sin2\\pi \\frac{y+k}n}=\\frac{(-1)^{(n-1)/2}n^2}{\\sin 2\\pi x+\\sin2\\pi y}$$ for all complex numbers $x$ and $y$ with $x+y,x-y-1/2\\not\\in\\mathbb Z$. We also determine the values of $\\prod_{k=1}^{(p-1)/2}(1+\\tan\\pi\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02155","kind":"arxiv","version":9},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.02155/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}