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Lean-Workbook Plus 47397: Trigonometric identity

Proved
WorkbookRestored.plus_47397

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

lean-workbookrestored-importssource-checked

For every real xx, sin⁡x(1−cos⁡x)1−cos⁡2x−sin⁡x(1+cos⁡x)1−cos⁡2x=−2cos⁡xsin⁡xsin⁡2x\frac{\sin x(1-\cos x)}{1-\cos^2x}-\frac{\sin x(1+\cos x)}{1-\cos^2x}=\frac{-2\cos x\sin x}{\sin^2x}. Lean’s total division includes vanishing denominators.

Source: Lean-Workbook row lean_workbook_plus_47397 (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_47397 :  ∀ x : ℝ, (sin x * (1 - cos x) / (1 - cos x ^ 2) - sin x * (1 + cos x) / (1 - cos x ^ 2) = -2 * cos x * sin x / sin x ^ 2)   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_47397; immutable original Prove2Me node 59ae3df8-68bd-4636-aaf2-8fbb98a4f4db

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