TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The complex Pythagorean identity by the identity theorem

Statement

For every complex number $z$ , $\sin^{2} z + \cos^{2} z = 1.$

This proof obtains the complex identity from its real restriction by the identity theorem.

Facts & Assumptions

Given: The complex sine and cosine of The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over $C$ and their algebra under sums and products (Linearity, product, reciprocal, and quotient rules for complex derivatives).

[L1]

For every real $x$ , $\sin^{2} x + \cos^{2} x = 1$ (Parity and the Pythagorean identity for sine and cosine).

[L2]

If two holomorphic functions on a complex domain agree on a set with an accumulation point in the domain, then they agree everywhere on the domain (Identity theorem for holomorphic functions).

[L3]

The functions $\sin$ , $\cos$ , $\sinh$ , and $\cosh$ are entire (Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives).

Proof

technique · direct

1.1

By [L3] and the holomorphic algebra laws, $h (z) : = \sin^{2} z + \cos^{2} z - 1$ is entire.

L3algebra

1.2

For every real $x$ , [L1] gives $h (x) = 0$ .

2.1

The real axis has accumulation point $0$ in the complex domain $C$ , so [L2] applied to $h$ and the zero function makes $h$ identically zero. Hence $\sin^{2} z + \cos^{2} z = 1$ for every complex $z$ .

step 1.1step 1.2L2∎

Remarks

This route is independent of the direct exponential-form calculation obtained by expanding the complex trigonometric dictionary and The addition formulas for complex trigonometric and hyperbolic functions. The proof above uses neither that addition formula nor its algebraic consequences.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, §2.4 (standard reference, not scraped)
B. V. Shabat, Introduction to Complex Analysis, §2.3 (standard reference, not scraped)