Double-angle and quadratic power-reduction identities

Depends on

• The addition formulas for sine and cosine
• Parity and the Pythagorean identity for sine and cosine
• Tangent, cotangent, secant, and cosecant on their exact natural domains

Used by

• Known-amplitude amplification with integer rounding Algorithm
• A fixed Grover iteration count can fail when a is unknown Counterexample
• A radial Poisson limit does not control a tangential path Counterexample
• A real-valued optimum is not a fractional Grover iteration Counterexample
• I₂(5) admits no crystallographic scaling and no reduced crystallographic root system with that base pairing Counterexample
• A cycle and an overlong arm: explicit non-positive witnesses Example
• A moved-space intersection in A₃ that is not the meet Example
• A prepared state with success amplitude one half Example
• Addition and half-angle identities compute the sine, cosine, and tangent of π/12 Example
• All meets and joins of the right weak order of A₂ (S₃), with the left order and the inversion sets compared Example
• An indefinite Coxeter form: infinite, but not of affine type Example
• B-tilde versus C-tilde: the n=2 coincidence and the duality behind the difference Example
• Bₙ and Cₙ define the same Coxeter diagram and the same Coxeter group Example
• Exact Grover rotation for four addresses and one marked item Example
• Exact sine and cosine values at π/10, π/5, and 2π/5 Example
• G2 from I2(6): the scaled realization and its twelve roots Example
• Gram determinants and principal minors of the non-crystallographic types H₃ and H₄ Example
• Morrie's law: cos(π/9)cos(2π/9)cos(4π/9)=1/8 Example
• Ordered roots and the mu-dot-root matrix in I2(5) Example
• Path determinants dₖ=dₖ₋₁-cos²(π/m)dₖ₋₂ and the three-arm inequality Example
• Poisson extension of an indicator arc Example
• Simple and reflection lengths of a long transposition in S₅ Example
• The A₂ root and weight lattices: P/Q has order three Example
• The Pontryagin dual of the circle is ℤ Example
• The Rayleigh quotient on an interval Example
• The solid generated by rotating y=sin x on [0,π] has volume π²/2 Example
• The two-dimensional Taylor–Green vortex on the flat torus Example
• The Wall form and line restrictions of a plane rotation, and the necessity of a common upper bound Example
• Two independent units in a real cubic field Example
• A Quantitative Logarithm Near the Identity in SU(2) Lemma
• Average Grover success for a uniform integer iteration count Lemma
• Cartan-number products, allowed edge labels, tree scalings and reflection stability Lemma
• Classical Poincare products for A, B, D and I2(m) from the permutation and signed-permutation models Lemma
• Constant BBHT stage success beyond the threshold scale Lemma
• Enumeration of the connected positive semidefinite corank-one diagrams Lemma
• Exact zero-label probability for counting Lemma
• Exceptional parabolic-orbit length certificates for E6, E7, E8, F4, H3 and H4 Lemma
• Finite amplification preserves the good state with a deterministic call cap Lemma
• Finite capped amplification succeeds with a supplied lower bound Lemma
• Periodic heat-stencil eigenvalues and θ amplification factor Lemma

…and 22 more results.

Dependency tree · two levels

10 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

• NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)