Prove2Me

Lean-Workbook Plus 31376: Trigonometric identity

Proved
WorkbookRestored.plus_31376

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

lean-workbookrestored-importssource-checked

For all real x,yx,y, cos⁡(arcsin⁡x)−1−x2=cos⁡(arcsin⁡y)−1−y2\cos(\arcsin x)-\sqrt{1-x^2}=\cos(\arcsin y)-\sqrt{1-y^2}. Lean’s total real inverse sine and square root are used, including outside [−1,1][-1,1].

Source: Lean-Workbook row lean_workbook_plus_31376 (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
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
open Real
Formal statement
theorem WorkbookRestored.plus_31376 x y : cos (arcsin x) - Real.sqrt (1 - x ^ 2) = cos (arcsin y) - Real.sqrt (1 - y ^ 2)   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_31376; immutable original Prove2Me node 268a6dd7-ac3b-4ee9-8c9c-fd76b0043167

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