Alphabeta MathHow statement and proof provenance work
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The subtraction formulas for sine and cosine
Statement
For all real , The conventions and prerequisite facts used below are recorded in The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine.
Facts & Assumptions
Given: Reals .
Proof
Apply the addition formulas to .
Replace and by their parity values and simplify.
Depends on
Used by
- A surjective map need not be a fibration Counterexample
- Addition and half-angle identities compute the sine, cosine, and tangent of π/12 Example
- Hopf circle fibration Example
- Mobius band as an interval bundle with monodromy Example
- Fourier uniqueness for continuous functions on the Euclidean torus Lemma
- Addition and subtraction formulas for tangent, cotangent, secant, and cosecant on their exact domains Theorem
- Product-to-sum and sum-to-product identities Theorem
Dependency tree · two levels
8 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)