Alphabeta MathFundamental Trigonometric Identities
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✓ 17 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 17 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Theorem
Pythagorean and parity identities for all six trigonometric functions on their natural domains
Theorem
Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions
Theorem
The subtraction formulas for sine and cosine
Theorem
Addition and subtraction formulas for tangent, cotangent, secant, and cosecant on their exact domains
Theorem
Double-angle and quadratic power-reduction identities
Theorem
Triple-angle identities for sine, cosine, and tangent
Theorem
Half-angle identities with the sign determined by the quadrant
Theorem
Product-to-sum and sum-to-product identities
Theorem
The tangent half-angle identities and rational parametrization of the unit circle away from (-1,0)
Theorem
t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle
Corollary
Every a cos x+b sin x has the amplitude-phase form R cos(x-φ)
Definition
Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials
Lemma
A real polynomial vanishing at a is divisible by x-a
Theorem
A nonzero real polynomial of degree n has no more than n distinct real roots
Definition
Chebyshev polynomials of the first and second kinds by their three-term recurrences
Lemma
Degrees and leading coefficients of the Chebyshev polynomials
Theorem
Tₙ(cosθ)=cos(nθ) and Uₙ(cosθ)sinθ=sin((n+1)θ) for every n∈ℕ
Theorem
Finite binomial formulas for cos(nθ) and sin(nθ)
Theorem