A cosine square in terms of the doubled angle

Proved

WorkbookRestored.plus_52

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

lean-workbookrestored-importssource-checked

For every real $x$ ,

\cos^2(2x)=\frac{1+\cos(4x)}2.

Formalization Note This restores source entry lean_workbook_plus_52 from the Apache-2.0 Lean-Workbook dataset. The mathematical proposition is unchanged; targeted imports and namespace openings supply the constants missing from the original Prove2Me node 8798f16e-9c7c-491b-95bb-6e4e35188d11. This is a separate corrected node because formal statements and preambles are immutable.

Preamble

import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Tactic
open Real

Formal statement

theorem WorkbookRestored.plus_52 : ∀ x : ℝ, Real.cos (2 * x) ^ 2 = (1 + Real.cos (4 * x)) / 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_52; original Prove2Me theorem ID 8798f16e-9c7c-491b-95bb-6e4e35188d11; Apache-2.0.

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