nyuuzyou/wikis huggingface dataset:
Well-Founded Relation is not necessarily Ordering

== Theorem ==
Let $\struct {S, \RR}$ be a relational structure.
Let $\RR$ be a well-founded relation on $S$.
Then it is not necessarily the case that $\RR$ is also either an ordering or a strict ordering.
== Proof ==
Proof by Counterexample:
Let $P$ be the set of all polynomials over $\R$ in one variable with real coefficients.
Let $\DD$ be a relation on $P$ defined as:
:$\forall p_0, p_1 \in P: \tuple {p_0, p_1} \in \DD$ $p_0$ is the derivative of $p_1$.
From Differentiation of Polynomials induces Well-Founded Relation, we have that $\DD$ is a well-founded relation on $P$.
Let $P_a$ be the polynomial defined as:
:$P_a = x^3$
Then the derivative of $P_a$ $x$ is:
:$P_a' = \dfrac \d {\d x} x^3 = 3 x^2$
Then the derivative of $P_a$ $x$ is:
:$P_a = \dfrac \d {\d x} 3 x^2 = 6 x$
So we have that:
:$\tuple {P_a', P_a} \in \DD$
and:
:$\tuple {P_a, P_a'} \in \DD$
but it is not the case that $\tuple {P_a, P_a} \in \DD$.
That is, $\DD$ is not transitive.
It follows that $\DD$ is neither an ordering nor a strict ordering.
== Sources ==
* : Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $7$
Category:Well-Founded Relations
Category:Orderings
Well-Orderable Set has Choice Function

== Theorem ==
Let $S$ be a well-orderable set.
Then $S$ has a choice function.
== Proof ==
Let $\preccurlyeq$ be a well-ordering on $S$.
Let $T \subseteq S$ be an arbitrary non-empty subset of $S$.
As $S$ is a well-ordered set, $T$ has a unique smallest element by $\preccurlyeq$.
Thus, we may define the choice function $C: \powerset S \setminus \set \O \to S$ as:
:$\forall T \in \powerset S \setminus \set \O: \map C T$ is the smallest element of $T$ under $\preccurlyeq$
This is the choice function we require.
== Sources ==
* : Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 4$ Well ordering and choice: Theorem $4.9$
Category:Choice Functions
Category:Well-Orderings
Well-Ordered Class is not Isomorphic to Initial Segment

== Theorem ==
Let $\struct {A, \preccurlyeq}$ be a well-ordered class.
There exists no order isomorphism from $\struct {A, \preccurlyeq}$ to an initial segment of $A$.
== Proof ==
Let $a \in A$.
Let $A_a$ be the initial segment of $A$ determined by $a$.
 $\phi: A \to A_a$ is an order isomorphism from $A$ to $A_a$.
From Order Automorphism on Well-Ordered Class is Forward Moving:
:$a \preccurlyeq \map \phi a$
But by definition of initial segment:
:$\map \phi a \notin A_a$
Hence $\phi$ cannot be an order isomorphism.
Hence by Proof by Contradiction there exists no such order isomorphism.
== Also presented as ==
== Sources ==
* : $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals: Theorem $1.7.4$
* : Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 2$ Isomorphisms of well orderings: Corollary $2.4$
Category:Well-Ordered Class is not Isomorphic to Initial Segment
Category:Order Isomorphisms
Category:Well-Orderings
Category:Initial Segments
Well-Ordered Class is not Isomorphic to Initial Segment/Also presented as

== Well-Ordered Class is not Isomorphic to Initial Segment: Also presented as ==
Some sources present this result in the context of well-ordered sets, and do not take into account the concept of classes.
The result is the same whether the domain of any hypothetical order isomorphism is a set or a propr class.
== Sources ==
* : $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals: Theorem $1.7.4$
Category:Well-Ordered Class is not Isomorphic to Initial Segment
Well-Ordered Classes are Isomorphic at Most Uniquely

== Theorem ==
Let $\struct {A, \preccurlyeq_A}$ and $\struct {B, \preccurlyeq_B}$ be well-ordered classes.
Then there exists at most one order isomorphism from $\struct {A, \preccurlyeq_A}$ to $\struct {B, \preccurlyeq_B}$
== Proof ==
Let $\phi$ and $\psi$ be order isomorphisms from $A$ to $B$.
Then from Inverse of Order Isomorphism is Order Isomorphism:
:the inverse mapping $\psi^{-1}$ is an order isomorphism from $B$ to $A$.
Hence from Composite of Order Isomorphisms is Order Isomorphism:
:$\phi \circ \psi^{-1}$ is an order isomorphism from $A$ to $A$.
But from Order Automorphism on Well-Ordered Class is Identity Mapping:
:$\phi \circ \psi^{-1}$ is the identity mapping.
From Composite of Mapping with Inverse of Another is Identity implies Mappings are Equal:
:$\phi = \psi$
== Sources ==
* : Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 2$ Isomorphisms of well orderings: Corollary $2.3$
Category:Order Isomorphisms
Category:Well-Orderings
Well-Ordered Induction

== Theorem ==
Let $\struct {A, \prec}$ be a strict well-ordering.
For all $x \in A$, let the $\prec$-initial segment of $x$ be a small class.
Let $B$ be a class such that $B \subseteq A$.
Let:
:$(1): \quad \forall x \in A: \paren {\paren {A \mathop \cap \map {\prec^{-1} } x} \subseteq B \implies x \in B}$
Then:
:$A = B$
That is, if a property passes from the initial segment of $x$ to $x$, then this property is true for all $x \in A$.
== Proof ==
 that $A \nsubseteq B$.
Then:
:$A \setminus B \ne 0$.
By Proper Well-Ordering Determines Smallest Elements, $A \setminus B$ must have some $\prec$-minimal element.
Thus:
:$\ds \exists x \in \paren {A \setminus B}: \paren {A \setminus B} \cap \map {\prec^{-1} } x = \O$
implies that:
:$A \cap \map {\prec^{-1} } x \subseteq B$
Hence this fulfils the hypothesis for $(1)$.
We have that $x \in A$.
so by $(1)$:
:$x \in B$
But this contradicts the fact that $x \in \paren {A \setminus B}$.
Thus by Proof by Contradiction:
:$A \subseteq B$
It follows by definition of set equality that:
:$A = B$
== Also see ==
* Well-Founded Induction shows that it is possible to weaken the hypotheses in order to drop the requirements that $\prec$ be well-ordering, replacing it with the requirement that $\prec$ be simply strictly well-founded (hence, the name well-founded induction) and to drop the requirement that the initial segments be sets (they may also be proper classes).
:It is important to note that such an approach involves the use of the axiom of foundation.
== Sources ==
* : $\S 6.27$
Category:Well-Orderings
Category:Class Theory
Category:Mathematical Induction
Well-Ordered Transitive Subset is Equal or Equal to Initial Segment

== Theorem ==
Let $\struct {\prec, A}$ be a well-ordered set.
For every $x \in A$, let every $\prec$-initial segment $A_x$ be a set.
Let $B$ be a subclass of $A$ such that
:$\forall x \in A: \forall y \in B: \paren {x \prec y \implies x \in B}$.
That is, $B$ must be $\prec$-transitive.
Then:
:$A = B$
or:
:$\exists x \in A: B = A_x$
== Proof ==
Let $A \ne B$.
Then $B \subsetneq A$.
Therefore, by Set Difference with Proper Subset:
:$A \setminus B \ne \O$
Then:
One direction of inclusion is proven.
By the hypothesis:
But $x \in A \land x \notin B$, so:
Therefore:
:$B \subseteq A_x$
and so 
:$B \subseteq A \cap A_x$
== Sources ==
* : $\S 7.45$
Category:Well-Orderings
Well-Ordering Minimal Elements are Unique

== Theorem ==
Let $\struct {S,\preceq}$ be a well-ordered set.
Then every non-empty subset of $S$ has a unique minimal element.
== Proof ==
The proof consists of a uniqueness and an existence part.
Let $S'$ be a non-empty subset of $S$.
=== Existence ===
As $S$ is well-ordered by $\preceq$, we have by definition the existence of a minimal elements $s$ of $S'$ (with respect to $\preceq$).
=== Uniqueness ===
Let $s, s'$ be minimal elements of $S'$. 
By Well-Ordering is Total Ordering, we have at least one of $s \preceq s'$, $s'\preceq s$.
Since both $s$ and $s'$ are minimal elements, we conclude that necessarily $s = s'$.
== Also see ==
* Definition:Well-Ordered Set
* Well-Ordering is Total Ordering
Category: Well-Orderings
Well-Ordering Principle

== Theorem ==
Every non-empty subset of $\N$ has a smallest (or first) element.
That is, the relational structure $\struct {\N, \le}$ on the set of natural numbers $\N$ under the usual ordering $\le$ forms a well-ordered set.
This is called the well-ordering principle.
=== Corollary ===
The well-ordering principle also holds for $\N_{\ne 0}$:
== Proof using Naturally Ordered Semigroup ==
== Proof using Von Neumann Construction ==
== Proof by Restriction of Real Numbers ==
== Also known as ==
This is otherwise known as:
* the well-ordering property (of $\N$)
* the least-integer principle
* the principle of the least element.
Note that some authors cite this as the well-ordering theorem.
However, this allows it to be confused even more easily with Zermelo's Well-Ordering Theorem, which states that any set can have an ordering under which that set is a well-ordered set.
== Also see ==
* Principle of Mathematical Induction
* Second Principle of Mathematical Induction
* Equivalence of Well-Ordering Principle and Induction
Some authors extend the scope of this theorem to include:
* Set of Integers Bounded Below has Smallest Element
* Set of Integers Bounded Above has Greatest Element
== Sources ==
* : Introduction $\S 4$: The natural numbers
* : $\S 13$: Arithmetic
* : $\S 2.3$: Theorem $2.16$
* : $\text {2-1}$ Euclid's Division Lemma: Exercise $2$
* : Chapter $1$: Properties of the Natural Numbers: $\S 20$
* : $\S 1.17$: Finite Induction and Well-Ordering for Positive Integers: $17.2$
* : $\S 4$: Number systems $\text{I}$: Peano's Axioms
* : Appendix $\text{A}.6$: Mathematical Induction
* : Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction
* : Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: $\mathbf{I}$
* : $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization
* : Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations
* : $\S 1.2.1$: Mathematical Induction: Exercise $15$
* : $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers
* 
Category:Number Theory
Category:Natural Numbers
Category:Named Theorems
Category:Well-Orderings
Category:Well-Ordering Principle
Category:Ordering on Natural Numbers
Well-Ordering Principle/Corollary

== Corollary to Well-Ordering Principle ==
Let $\le_1$ denote the restriction of the usual ordering $\le$ to the set $\N_{\ne 0}$ of natural numbers without zero.
The relational structure $\struct {\N_{\ne 0}, \le_1}$ forms a well-ordered set.
== Proof ==
We have by the Well-Ordering Principle that $\struct {\N, \le}$ forms a well-ordered set.
We also have that $\N_{\ne 0}$ is a subset of $\N$.
The result follows from Subset of Well-Ordered Set is Well-Ordered.
== Sources ==
* : Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations
Category:Well-Ordering Principle
Well-Ordering Principle/Proof by Restriction of Real Numbers

== Theorem ==
== Proof ==
Let $S$ be a non-empty subset of the set of natural numbers $\N$.
We take as axiomatic that $\N$ is itself a subset of the set of real numbers $\R$.
Thus $S \subseteq \R$.
By definition:
:$\forall n \in \N: n \ge 0$
and so:
:$\forall n \in S: n \ge 0$
Hence $0$ is a lower bound of $S$.
This establishes the fact that $S$ is bounded below.
By the Continuum Property, we have that $S$ admits an infimum.
Hence let $b = \inf S$.
Because $b$ is the infimum of $S$, it follows that $b + 1$ is not a lower bound of $S$.
So, for some $n \in S$:
:$n < b + 1$
 $n$ is not the smallest element of $S$.
Then there exists $m \in S$ such that:
:$b \le m < n < b + 1$
from which it follows that:
:$0 < n - m < 1$
But there exist no natural numbers $k$ such that $0 < k < 1$.
Hence $n = b$ is the smallest element of $S$.
== Sources ==
* : $\S 3$: Natural Numbers: $\S 3.5$
Category:Well-Ordering Principle
Well-Ordering Principle/Proof using Naturally Ordered Semigroup

== Theorem ==
== Proof ==
Consider the natural numbers $\N$ defined as the naturally ordered semigroup $\struct {S, \circ, \preceq}$.
From its definition, $\struct {S, \circ, \preceq}$ is well-ordered by $\preceq$.
The result follows.
As $\N_{\ne 0} = \N \setminus \set 0$, by Set Difference is Subset $\N_{\ne 0} \subseteq \N$.
As $\N$ is well-ordered, by definition, every subset of $\N$ has a smallest element.
Category:Well-Ordering Principle
Well-Ordering Principle/Proof using Von Neumann Construction

== Theorem ==
== Proof ==
From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is a minimally inductive class under the successor mapping.
From Successor Mapping on Natural Numbers is Progressing, this successor mapping is a progressing mapping.
The result is a direct application of Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation.
== Sources ==
* : Chapter $3$: The Natural Numbers: $\S 5$ Applications to natural numbers: Theorem $5.5$
Category:Well-Ordering Principle
Well-Ordering is not necessarily Usual Ordering

== Theorem ==
Let $S$ be a set of numbers.
According to Zermelo's Well-Ordering Theorem, $S$ can be well-ordered.
However, the usual ordering on $S$ may not necessarily be a well-ordering.
== Proof ==
From Rational Numbers are Well-Orderable, it is possible to apply a well-ordering to the set of rational numbers $\Q$.
However, the usual ordering on $\Q$ is not a well-ordering.
Indeed:
:$\set {x \in \Q: x \le 0}$
has no smallest element.
== Sources ==
* : Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Discussion
Category:Well-Orderings
Well-Ordering of Class of All Ordinals under Subset Relation

== Theorem ==
Let $\On$ denote the class of all ordinals.
$\On$ is well-ordered by the subset relation such that the following $3$ conditions hold:
== Proof ==
We have that Class of All Ordinals is $g$-Tower.
By Zero is Smallest Ordinal, $0$ is the smallest element of $\On$.
We identify the natural number $0$ via the von Neumann construction of the natural numbers as:
:$0 := \O$
The result then follows directly from $g$-Tower is Well-Ordered under Subset Relation.
== Sources ==
* : Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.11$
Category:Class of All Ordinals
Category:Well-Orderings
Well-Ordering on Class is not necessarily Proper

== Theorem ==
Let $A$ be a class.
Let $\preccurlyeq$ be a well-ordering on $A$.
Then it is not necessarily the case that $\preccurlyeq$ is a proper well-ordering.
== Proof ==
Proof by Counterexample:
== Sources ==
* : Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries: Exercise $1.2$
Category:Well-Orderings
Category:Proper Well-Orderings
Well-Ordering on Set is Proper Well-Ordering

== Theorem ==
Let $\struct {S, \preccurlyeq}$ be a well-ordered set.
Then $\preccurlyeq$ is a proper well-ordering.
== Proof ==
By definition, a proper well-ordering is a well-ordering on a class such that:
:every proper lower section of $S$ is a set.
We have that a proper lower section of $S$ is a subclass of $S$.
But here we have that $S$ is a set.
The result follows from Subclass of Set is Set.
== Sources ==
* : Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries
Category:Proper Well-Orderings
Well Inside Elements Form Filter

== Theorem ==
Let $L = \struct{S, \vee, \wedge, \preceq}$ be a distributive lattice with greatest element $\top$ and smallest element $\bot$.
Let $\eqslantless$ denote the well inside relation on $L$.
Then:
:$\forall a \in S : \set{b \in S: a \eqslantless b}$ is a lattice filter
== Proof ==
Let $a \in S$.
Let $F = \set{b \in S: a \eqslantless b}$.
=== $F$ is an Upper Section ===
Let $b \in F$ and $c \in S : b \preceq c$.
By :
:$a \preceq a$
Hence we have:
:$a \preceq a \eqslantless b \preceq c$
From Well Inside Relation Extends to Predecessor and Successor:
:$a \eqslantless c$
Hence:
:$c \in F$
It follows that $F$ is an upper section by definition.
=== $F$ is a Meet Subsemilattice ===
Let $b, c \in F$.
By definition of well inside relation:
:$\exists x, y \in S : a \wedge x = \bot, b \vee x = \top, a \wedge y = \bot, c \vee y = \top$ 
We have:
Also we have:
By definition of well inside relation:
:$a \eqslantless b \wedge c$
By definition of $F$:
:$b \wedge c \in F$
It follows that $F$ is an meet subsemilattice by definition.
It follows that $F$ is a lattice filter by definition.
The result follows.
== Sources ==
* : Chapter $\text {III}$: Compact Hausdorff Spaces, $\S1.1$
Category:Well Inside Relation
Well Inside Implies Predecessor

== Theorem ==
Let $L = \struct{S, \vee, \wedge, \preceq}$ be a distributive lattice with greatest element $\top$ and smallest element $\bot$.
Let $\eqslantless$ denote the well inside relation on $L$.
Then:
:$\forall a,b \in S : a \eqslantless b \implies a \preceq b$
== Proof ==
Let $a, b \in S : a \eqslantless b$.
By definition of well inside relation:
:$\exists c \in S : a \wedge c = \bot, b \vee c = \top$
We have:
From Predecessor is Infimum:
:$a \preceq b$
The result follows.
== Sources ==
* : Chapter $\text {III}$: Compact Hausdorff Spaces, $\S1.1$
Category:Well Inside Relation
Well Inside Relation Extends to Predecessor and Successor

== Theorem ==
Let $L = \struct{S, \vee, \wedge, \preceq}$ be a distributive lattice with greatest element $\top$ and smallest element $\bot$.
Let $\eqslantless$ denote the well inside relation on $L$.
Then:
:$\forall a,b,c,d \in S : a \preceq b \eqslantless c \preceq d \implies a \eqslantless d$
== Proof ==
Let $a,b,c,d \in S : a \preceq b \eqslantless c \preceq d$
By definition of well inside relation:
:$\exists x \in S : b \wedge x = \bot, c \vee x = \top$
We have:
From :
:$a \wedge x = \bot$
Similarly, we have:
From :
:$d \vee x = \top$
By definition of well inside relation:
:$a \eqslantless d$
The result follows.
== Sources ==
* : Chapter $\text {III}$: Compact Hausdorff Spaces, $\S1.1$
Category:Well Inside Relation
Werner Formula for Cosine by Cosine/Also presented as

== Werner Formula for Cosine by Cosine: Also presented as ==
The Werner Formula for Cosine by Cosine can also be seen in the form:
:$2 \cos \alpha \cos \beta = \map \cos {\alpha - \beta} + \map \cos {\alpha + \beta}$
== Sources ==
* : $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
* 
Category:Werner Formula for Cosine by Cosine
Werner Formula for Cosine by Cosine/Examples/2 Cosine 20 Cosine 50

== Example of Use of Werner Formula for Cosine by Cosine ==
:$2 \cos 20 \degrees \cos 50 \degrees = \cos 30 \degrees + \cos 70 \degrees$
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Exercises $\text {XXXII}$: $8$.
Category:Werner Formula for Cosine by Cosine
Werner Formula for Cosine by Sine/Also presented as

== Werner Formula for Cosine by Sine: Also presented as ==
The Werner Formula for Cosine by Sine can also be seen in the form:
:$2 \cos \alpha \sin \beta = \map \sin {\alpha + \beta} - \map \sin {\alpha - \beta}$
== Sources ==
* 
Category:Werner Formula for Cosine by Sine
Werner Formula for Cosine by Sine/Examples/2 Cosine 2 A Sine 4 A

== Example of Use of Werner Formula for Cosine by Sine ==
:$2 \cos 2 A \sin 4 A = \sin 6 A + \sin 2 A$
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Exercises $\text {XXXII}$: $12$.
Category:Werner Formula for Cosine by Sine
Werner Formula for Cosine by Sine/Examples/2 Cosine 30 Sine 20

== Example of Use of Werner Formula for Cosine by Sine ==
:$2 \cos 30 \degrees \sin 20 \degrees = \sin 50 \degrees - \sin 10 \degrees$
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Exercises $\text {XXXII}$: $9$.
Category:Werner Formula for Cosine by Sine
Werner Formula for Sine by Cosine/Also presented as

== Werner Formula for Sine by Cosine: Also presented as ==
The Werner Formula for Sine by Cosine can also be seen in the form:
:$2 \sin \alpha \cos \beta = \map \sin {\alpha - \beta} + \map \sin {\alpha + \beta}$
== Sources ==
* : $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
* 
Category:Werner Formula for Sine by Cosine
Werner Formula for Sine by Cosine/Examples/2 Sine 30 Cosine 10

== Example of Use of Werner Formula for Sine by Cosine ==
:$2 \sin 30 \degrees \cos 10 \degrees = \sin 40 \degrees + \sin 20 \degrees$
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Exercises $\text {XXXII}$: $10$.
Category:Werner Formula for Sine by Cosine
Werner Formula for Sine by Sine/Also presented as

== Werner Formula for Sine by Sine: Also presented as ==
The Werner Formula for Sine by Sine can also be seen in the form:
:$2 \sin \alpha \sin \beta = \map \cos {\alpha - \beta} - \map \cos {\alpha + \beta}$
== Sources ==
* : $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
* 
Category:Werner Formula for Sine by Sine
Werner Formula for Sine by Sine/Examples/2 Sine 10 Sine 30

== Example of Use of Werner Formula for Sine by Sine ==
:$2 \sin 10 \degrees \sin 30 \degrees = \cos 20 \degrees - \cos 40 \degrees$
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Exercises $\text {XXXII}$: $7$.
Category:Werner Formula for Sine by Sine
Werner Formula for Sine by Sine/Examples/2 Sine 3 A Sine 5 A

== Example of Use of Werner Formula for Sine by Sine ==
:$2 \sin 3 A \sin 5 A = \cos 2 A - \cos 8 A$
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Exercises $\text {XXXII}$: $11$.
Category:Werner Formula for Sine by Sine
Werner Formulas

== Theorem ==
=== Cosine by Cosine ===
=== Sine by Sine ===
=== Sine by Cosine ===
=== Cosine by Sine ===
== Hyperbolic Functions ==
While did not consider the hyperbolic functions, it is convenient to use his name to identify them, as follows:
=== Hyperbolic Cosine by Hyperbolic Cosine ===
=== Hyperbolic Sine by Hyperbolic Sine ===
=== Hyperbolic Sine by Hyperbolic Cosine ===
== Also presented as ==
== Also known as ==
== Also see ==
* Prosthaphaeresis Formulas
== Sources ==
* 
* 
Category:Werner Formulas
Category:Trigonometric Identities
Category:Hyperbolic Functions
Werner Formulas/Also known as

== Werner Formulas: Also known as ==
The Werner formulas are also known as the product formulas or product formulae.
They can also be called the product-to-sum formulas.
Some sources call them Simpson's formulas, but this appears not to be backed up by the literature, and this name is usually applied to another set of formulas altogether.
== Sources ==
* 
* 
* : Appendix $12$: Trigonometric formulae: Product formulae
* : Appendix $14$: Trigonometric formulae: Product formulae
Category:Werner Formulas
Werner Formulas/Cosine by Cosine

== Theorem ==
:$\cos \alpha \cos \beta = \dfrac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2$
where $\cos$ denotes cosine.
== Proof ==
== Also presented as ==
== Examples ==
== Also see ==
* Werner Formula for Sine by Sine
* Werner Formula for Sine by Cosine
* Werner Formula for Cosine by Sine
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.66$
* 
* 
* : Appendix $12$: Trigonometric formulae: Product formulae
* : Appendix $14$: Trigonometric formulae: Product formulae
Category:Werner Formula for Cosine by Cosine
Category:Werner Formulas
Category:Cosine Function
Werner Formulas/Cosine by Sine

== Theorem ==
:$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$
where $\cos$ denotes cosine and $\sin$ denotes sine.
== Proof 1 ==
== Proof 2 ==
== Also presented as ==
== Examples ==
== Also see ==
* Werner Formula for Sine by Sine
* Werner Formula for Cosine by Cosine
* Werner Formula for Sine by Cosine
== Sources ==
* 
* 
* : Appendix $12$: Trigonometric formulae: Product formulae
* : Appendix $14$: Trigonometric formulae: Product formulae
Category:Werner Formula for Cosine by Sine
Category:Werner Formulas
Category:Sine Function
Category:Cosine Function
Werner Formulas/Hyperbolic Cosine by Hyperbolic Cosine

== Theorem ==
:$\cosh x \cosh y = \dfrac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$
where $\cosh$ denotes hyperbolic cosine.
== Proof ==
== Also presented as ==
This result can also be seen presented as:
:$2 \cosh x \cosh y = \cosh \paren {x + y} + \cosh \paren {x - y}$
== Sources ==
* : $\S 4$. Elementary Functions of a Complex Variable: Exercise $8 \ \text{(i)}$
* : $\S 8$: Hyperbolic Functions: $8.47$: Sum, Difference and Product of Hyperbolic Functions
Category:Werner Formulas
Category:Hyperbolic Cosine Function