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Lean-Workbook Plus 19576: Trigonometric inequality

Proved
WorkbookRestored.plus_19576

by wamlart · Sep 12, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

lean-workbookrestored-importssource-checked

If tan⁡x<0\tan x<0, then sin⁡x\sin x and cos⁡x\cos x are both nonzero and sin⁡xcos⁡x<0\sin x\cos x<0.

Source: Lean-Workbook row lean_workbook_plus_19576 (Apache-2.0). The unchanged proposition comes from the original record; its missing mathematical imports and namespaces are restored here.

Preamble
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
open Real
Formal statement
theorem WorkbookRestored.plus_19576 (x : ℝ) (hx : tan x < 0) : (sin x ≠ 0 ∧ cos x ≠ 0 ∧ sin x * cos x < 0)   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_19576; immutable original Prove2Me node 13dfb87a-5664-4692-83fb-506aa6c89f0e

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