hcju/proofwikips huggingface dataset:
Pasch's Theorem
Let $a, b, c, d$ be points on a line. Let $\left({a, b, c}\right)$ denote that $b$ lies between $a$ and $c$. Then $\left({a, b, c}\right)$ and $\left({b, c, d}\right)$ together imply that $\left({a, b, d}\right)$. That is, if: : $b$ is between $a$ and $c$ and: : $c$ is between $b$ and $d$ then: : $b$ is between $a$ and $d$.
Forward Difference of Power
:$\Delta c^x = \paren {c - 1} c^x$ where $\Delta$ denotes the forward difference operator.
Forward Difference of Harmonic Number Function
Let $H_x$ denote the harmonic number function. Then: :$\Delta H_x = \dfrac 1 {x + 1}$ where $\Delta H_x$ denotes the forward difference operator.
Equality of Ordered Tuples
Let $a = \tuple {a_1, a_2, \ldots, a_n}$ and $b = \tuple {b_1, b_2, \ldots, b_n}$ be ordered tuples. Then: :$a = b \iff \forall i: 1 \le i \le n: a_i = b_i$ That is, for two ordered tuples to be equal, all the corresponding elements have to be equal.
Zero of Power Set with Intersection
Let $S$ be a set and let $\powerset S$ be its power set. Consider the algebraic structure $\struct {\powerset S, \cap}$, where $\cap$ denotes set intersection. Then the empty set $\O$ serves as the zero element for $\struct {\powerset S, \cap}$.
Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group
Let $\R_{>0}$ be the set of strictly positive real numbers: :$\R_{>0} = \set {x \in \R: x > 0}$ The structure $\struct {\R_{>0}, \times}$ is an uncountable abelian group.
Law of Mass Action
Let $\AA$ and $\BB$ be two chemical substances in a solution $C$ which are involved in a second-order reaction. Let $x$ grams of $\CC$ contain $a x$ grams of $\AA$ and $b x$ grams of $\BB$, where $a + b = 1$. Let there be $a A$ grams of $\AA$ and $b B$ grams of $\BB$ at time $t = t_0$, at which time $x = 0$. Then: :$x = \begin{cases} \dfrac {k A^2 a b t} {k A a b t + 1} & : A = B \\ & \\ \dfrac {A B e^{-k \paren {A - B} a b t}} {A - B e^{-k \paren {A - B} a b t}} & : A \ne B \end{cases}$ This is known as the '''law of mass action'''
Motion of Pendulum
Consider a pendulum consisting of a bob whose mass is $m$, at the end of a rod of negligible mass of length $a$. Let the bob be pulled to one side so that the rod is at an angle $\alpha$ from the vertical and then released. Let $T$ be the time period of the pendulum, that is, the time through which the bob takes to travel from one end of its path to the other, and back again. Then: :$T = 2 \sqrt {\dfrac a g} \map K k$ where: :$k = \map \sin {\dfrac \alpha 2}$ :$\map K k$ is the complete elliptic integral of the first kind.
Body under Constant Acceleration
=== $(1):$ Velocity after Time === {{:Body under Constant Acceleration/Velocity after Time}} === $(2):$ Distance after Time === {{:Body under Constant Acceleration/Distance after Time}} === $(3):$ Velocity after Distance === {{:Body under Constant Acceleration/Velocity after Distance}} where: :$\mathbf u$ is the velocity at time $t = 0$ :$\mathbf v$ is the velocity at time $t$ :$\mathbf s$ is the displacement of $B$ from its initial position at time $t$ :$\cdot$ denotes the dot product.
One-Parameter Family of Curves for First Order ODE
Every one-parameter family of curves is the general solution of some first order ordinary differential equation. Conversely, every first order ordinary differential equation has as its general solution some one-parameter family of curves.
Orthogonal Trajectories of One-Parameter Family of Curves
Every one-parameter family of curves has a unique family of orthogonal trajectories.
Brachistochrone is Cycloid
The shape of the brachistochrone is a cycloid.
Equation of Cycloid
Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \tuple {x, y}$ is described by the equations: :$x = a \paren {\theta - \sin \theta}$ :$y = a \paren {1 - \cos \theta}$
Length of Arc of Cycloid
Let $C$ be a cycloid generated by the equations: :$x = a \paren {\theta - \sin \theta}$ :$y = a \paren {1 - \cos \theta}$ Then the length of one arc of the cycloid is $8 a$.
Goldbach's Weak Conjecture
Every odd integer greater than $7$ is the sum of three odd primes.
Goldbach Conjecture implies Weak Goldbach Conjecture
The Goldbach Conjecture: : Every even integer greater than $2$ is the sum of two primes implies Goldbach's Weak Conjecture: : Every odd integer greater than $7$ is the sum of three odd primes.
Unique Factorization Theorem
Let $\struct {D, +, \times}$ be a Euclidean domain. Then $\struct {D, +, \times}$ is a unique factorization domain.
Integral of Power
:$\displaystyle \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$
Volume of Sphere
The volume $V$ of a sphere of radius $r$ is given by: :$V = \dfrac {4 \pi r^3} 3$
Einstein's Mass-Energy Equation
The energy imparted to a body to cause that body to move causes the body to increase in mass by a value $M$ as given by the equation: :$E = M c^2$ where $c$ is the speed of light.
Dimension of Gravitational Constant
The dimension of the gravitational constant $G$ is $M^{-1} L^3 T^{-2}$.
Dimension of Spring Force Constant
The dimension of a spring force constant is $\mathsf {M T}^{-2}$.
Vieta's Formula for Pi
:$\pi = 2 \times \dfrac 2 {\sqrt 2} \times \dfrac 2 {\sqrt {2 + \sqrt 2} } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt 2} } } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt 2 } } } } \times \cdots$
Derivative of Arc Length
Let $C$ be a curve in the cartesian plane described by the equation $y = \map f x$. Let $s$ be the length along the arc of the curve from some reference point $P$. Then the derivative of $s$ with respect to $x$ is given by: :$\dfrac {\d s} {\d x} = \sqrt {1 + \paren {\dfrac {\d y} {\d x} }^2}$
Duality Principle for Sets
Any identity in set theory which uses any or all of the operations: :Set intersection $\cap$ :Set union $\cup$ :Empty set $\O$ :Universal set $\mathbb U$ and none other, remains valid if: :$\cap$ and $\cup$ are exchanged throughout :$\O$ and $\mathbb U$ are exchanged throughout.
Cartesian Product is not Associative
Let $A, B, C$ be non-empty sets. Then: :$A \times \paren {B \times C} \ne \paren {A \times B} \times C$ where $A \times B$ is the cartesian product of $A$ and $B$.
Rational Addition is Commutative
The operation of addition on the set of rational numbers $\Q$ is commutative: :$\forall x, y \in \Q: x + y = y + x$
Rational Addition is Associative
The operation of addition on the set of rational numbers $\Q$ is associative: :$\forall x, y, z \in \Q: x + \paren {y + z} = \paren {x + y} + z$
Subtraction on Numbers is Anticommutative
The operation of subtraction on the numbers is anticommutative. That is: :$a - b = b - a \iff a = b$
Commutative Law of Addition
:$\forall x, y \in \mathbb F: x + y = y + x$ That is, the operation of addition on the standard number sets is commutative.
Integer Addition is Commutative
The operation of addition on the set of integers $\Z$ is commutative: :$\forall x, y \in \Z: x + y = y + x$
Integer Addition is Associative
The operation of addition on the set of integers $\Z$ is associative: :$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$
Natural Number Addition is Commutative
The operation of addition on the set of natural numbers $\N$ is commutative: :$\forall m, n \in \N: m + n = n + m$
Associative Law of Addition
:$\forall x, y, z \in \mathbb F: x + \left({y + z}\right) = \left({x + y}\right) + z$ That is, the operation of addition on the standard number sets is associative.
Natural Number Addition is Associative
The operation of addition on the set of natural numbers $\N$ is associative: :$\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$
Rational Multiplication is Commutative
The operation of multiplication on the set of rational numbers $\Q$ is commutative: :$\forall x, y \in \Q: x \times y = y \times x$
Rational Multiplication is Associative
The operation of multiplication on the set of rational numbers $\Q$ is associative: :$\forall x, y, z \in \Q: x \times \paren {y \times z} = \paren {x \times y} \times z$
Natural Number Multiplication is Commutative
The operation of multiplication on the set of natural numbers $\N$ is commutative: :$\forall x, y \in \N: x \times y = y \times x$
Natural Number Multiplication is Associative
The operation of multiplication on the set of natural numbers $\N$ is associative: :$\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$
Natural Number Multiplication Distributes over Addition
The operation of multiplication is distributive over addition on the set of natural numbers $\N$: :$\forall x, y, z \in \N:$ ::$\paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$ ::$z \times \paren {x + y} = \paren {z \times x} + \paren {z \times y}$
Multiplication of Numbers Distributes over Addition
On all the number systems: :natural numbers $\N$ :integers $\Z$ :rational numbers $\Q$ :real numbers $\R$ :complex numbers $\C$ the operation of multiplication is distributive over addition: :$m \paren {n + p} = m n + m p$ :$\paren {m + n} p = m p + n p$
Rational Multiplication Distributes over Addition
The operation of multiplication on the set of rational numbers $\Q$ is distributive over addition: :$\forall x, y, z \in \Q: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$ :$\forall x, y, z \in \Q: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$
Associative Law of Multiplication
:$\forall x, y, z \in \mathbb F: x \times \paren {y \times z} = \paren {x \times y} \times z$ That is, the operation of multiplication on the standard number sets is associative.
Commutative Law of Multiplication
:$\forall x, y \in \mathbb F: x + y = y + x$ That is, the operation of multiplication on the standard number sets is commutative.
Subtraction on Numbers is Not Associative
The operation of subtraction on the numbers is not associative. That is, in general: :$a - \paren {b - c} \ne \paren {a - b} - c$
Identity Element of Multiplication on Numbers
On all the number systems: * natural numbers $\N$ * integers $\Z$ * rational numbers $\Q$ * real numbers $\R$ * complex numbers $\C$ the identity element of multiplication is one ($1$).
Identity Element of Addition on Numbers
On all the number systems: * natural numbers $\N$ * integers $\Z$ * rational numbers $\Q$ * real numbers $\R$ * complex numbers $\C$ the identity element of addition is zero ($0$).
Zero Element is Unique
Let $\struct {S, \circ}$ be an algebraic structure that has a zero element $z \in S$. Then $z$ is unique.
Group with Zero Element is Trivial
Let $\struct {G, \circ}$ be a group. Let $\struct {G, \circ}$ have a zero element. Then $\struct {G, \circ}$ is the trivial group.
Count of Binary Operations on Set
Let $S$ be a set whose cardinality is $n$. The number $N$ of different binary operations that can be applied to $S$ is given by: :$N = n^{\paren {n^2} }$
Count of Commutative Binary Operations on Set
Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different commutative binary operations that can be applied to $S$ is given by: :$N = n^{\frac {n \paren {n + 1} } 2}$
Count of Binary Operations with Fixed Identity
Let $S$ be a set whose cardinality is $n$. Let $x \in S$. The number $N$ of possible different binary operations such that $x$ is an identity element that can be applied to $S$ is given by: :$N = n^{\left({\left({n-1}\right)^2}\right)}$
Count of Commutative Binary Operations with Fixed Identity
Let $S$ be a set whose cardinality is $n$. Let $x \in S$. The number $N$ of possible different commutative binary operations such that $x$ is an identity element that can be applied to $S$ is given by: :$N = n^{\frac {n \left({n-1}\right)}2}$
Count of Binary Operations with Identity
Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different binary operations which have an identity element that can be applied to $S$ is given by: :$N = n^{\paren {n - 1}^2 + 1}$
Power of Element in Subgroup
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\struct {H, \circ}$ be a subgroup of $\struct {G, \circ}$. Let $x \in H$. Then: :$\forall n \in \Z: x^n \in H$
Element to Power of Remainder
:$\forall n \in \Z: n = q k + r: 0 \le r < k \iff a^n = a^r$
Element to Power of Multiple of Order is Identity
:$\forall n \in \Z: k \divides n \iff a^n = e$
List of Elements in Finite Cyclic Group
:$\set {a^0, a^1, a^2, \ldots, a^{k - 1} }$ is a complete repetition-free list of the elements of $\gen a$
Rising Sum of Binomial Coefficients
:$\displaystyle \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1} = \binom {n + m + 1} m$
Sum of Even Index Binomial Coefficients
:$\displaystyle \sum_{i \mathop \ge 0} \binom n {2 i} = 2^{n - 1}$
Sum of Odd Index Binomial Coefficients
:$\displaystyle \sum_{i \mathop \ge 0} \binom n {2 i + 1} = 2^{n - 1}$
Increasing Alternating Sum of Binomial Coefficients
:$\displaystyle \sum_{j \mathop = 0}^n \left({-1}\right)^{n + 1} j \binom n j = 0$
Sum of Squares of Binomial Coefficients
:$\displaystyle \sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces
Let $\mathbb K$ be a subfield of $\C$. Let $V$ be a semi-inner product space over $\mathbb K$. Let $x, y$ be vectors in $V$. Then: :$\size {\innerprod x y}^2 \le \innerprod x x \innerprod y y$
Cauchy-Bunyakovsky-Schwarz Inequality/Definite Integrals
Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$. Then: :$\displaystyle \paren {\int_a^b \map f t \, \map g t \rd t}^2 \le \int_a^b \paren {\map f t}^2 \rd t \int_a^b \paren {\map g t}^2 \rd t$
Cauchy-Schwarz Inequality/Complex Numbers
:$\displaystyle \paren {\sum \cmod {w_i}^2} \paren {\sum \cmod {z_i}^2} \ge \cmod {\sum w_i z_i}^2$ where all of $w_i, z_i \in \C$.
Ceiling of Floor is Floor
:$\left \lceil {\left \lfloor {x}\right \rfloor}\right \rceil = \left \lfloor {x}\right \rfloor$
Floor of Ceiling is Ceiling
: $\left \lfloor {\left \lceil {x}\right \rceil}\right \rfloor = \left \lceil {x}\right \rceil$
Ceiling of Negative equals Negative of Floor
:$\ceiling {-x} = -\floor x$
Floor of Negative equals Negative of Ceiling
:$\left \lfloor {-x}\right \rfloor = - \left \lceil {x}\right \rceil$
Integer is Coprime to 1
Every integer is coprime to $1$. That is: :$\forall n \in \Z: n \perp 1$
Integers Coprime to Zero
The only integers which are coprime to zero are $1$ and $-1$. That is: :$n \in \Z: n \perp 0 \iff n \in \set {1, -1}$ In particular, note that two integers which are coprime to each other cannot both be $0$.
Sum of Binomial Coefficients over Upper Index
{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^n \binom j m       | r = \binom {n + 1} {m + 1}       | c =  }} {{eqn | r = \dbinom 0 m + \dbinom 1 m + \dbinom 2 m + \cdots + \dbinom n m = \dbinom {n + 1} {m + 1}       | c =  }} {{end-eqn}}
Divisibility of Product of Consecutive Integers
The product of $n$ consecutive positive integers is divisible by the product of the ''first'' $n$ consecutive positive integers. That is: :$\displaystyle \forall m, n \in \Z_{>0}: \exists r \in \Z: \prod_{k \mathop = 1}^n \paren {m + k} = r \prod_{k \mathop = 1}^n k$
Equal Numbers are Congruent
:$\forall x, y, z \in \R: x = y \implies x \equiv y \pmod z$ where $x \equiv y \pmod z$ denotes congruence modulo $z$.
Complex Roots of Unity in Exponential Form
Let $n \in \Z$ be an integer such that $n > 0$. Let $z \in \C$ be a complex number such that $z^n = 1$. Then: :$U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$ where $U_n$ is the set of $n$th roots of unity. That is: :$z \in \set {1, e^{2 i \pi / n}, e^{4 i \pi / n}, \ldots, e^{2 \paren {n - 1} i \pi / n} }$ Thus for every integer $n$, the number of $n$th roots of unity is $n$. Setting $\omega := e^{2 i \pi / n}$, $U_n$ can then be written as: :$U_n = \set {1, \omega, \omega^2, \ldots, \omega^{n - 1} }$
Roots of Unity under Multiplication form Cyclic Group
Let $n \in \Z$ be an integer such that $n > 0$. The $n$th complex roots of unity under the operation of multiplication form the cyclic group which is isomorphic to $C_n$.
Inverse Mapping is Unique
Let $f: S \to T$ be a mapping. If $f$ has an inverse mapping, then that inverse mapping is unique. That is, if: :$f$ and $g$ are inverse mappings of each other and :$f$ and $h$ are inverse mappings of each other then $g = h$.
Special Linear Group is Subgroup of General Linear Group
Let $K$ be a field whose zero is $0_K$ and unity is $1_K$. Let $\SL {n, K}$ be the special linear group of order $n$ over $K$. Then $\SL {n, K}$ is a subgroup of the general linear group $\GL {n, K}$.
Determinant of Unit Matrix
Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. The determinant of the unit matrix of order $n$ over $R$ is equal to $1_R$.
Determinant of Inverse Matrix
Let $K$ be a field whose zero is $0_K$ and whose unity is $1_K$. Let $\mathbf A$ be an invertible matrix of order $n$ over $K$. Then the determinant of its inverse is given by: :$\map \det {\mathbf A^{-1} } = \dfrac {1_K} {\map \det {\mathbf A} }$
Transposition is of Odd Parity
Let $S_n$ denote the set of permutations on $n$ letters. Let $\pi \in S_n$ be a transposition. Then $\pi$ is of odd parity.
Subset Product Action is Group Action
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\powerset G$ be the power set of $\struct {G, \circ}$. For any $S \in \powerset G$ and for any $g \in G$, the subset product action: :$\forall g \in G: \forall S \in \powerset G: g * S = g \circ S$ is a group action.
Subgroup of Elements whose Order Divides Integer
Let $A$ be an abelian group. Let $k \in \Z$ and $B$ be a set of the form: :$\left\{{x \in A : x^k = e}\right\}$ Then $B$ is a subgroup of $A$.
Abelian Group Factored by Prime
Let $G$ be a finite abelian group. Let $p$ be a prime. Factor the order of $G$ as: :$\order G = m p^n$ such that $p$ does not divide $m$. Then: :$G = H \times K$ where: :$H = \set {x \in G : x^{p^n} = e}$ and: :$K = \set {x \in G : x^m = e}$ Furthermore: :$\order H = p^n$
Abelian Group of Prime-power Order is Product of Cyclic Groups
Let $G$ be an abelian group of prime-power order. Let $a$ be an element of maximal order in $G$. Then $G$ can be written in the form $\gen a \times K$ for some $K \le G$.
Power of Product in Abelian Group
Let $\struct {G, \circ}$ be an abelian group. Then: :$\forall x, y \in G: \forall k \in \Z: \paren {x \circ y}^k = x^k \circ y^k$
Identity Element of Natural Number Addition is Zero
The identity element for the natural numbers under addition is zero ($0$): :$\forall n \in \N: 0 + n = n$
Simpson's Formulas
=== Cosine by Cosine === {{:Simpson's Formulas/Cosine by Cosine}} === Sine by Sine === {{:Simpson's Formulas/Sine by Sine}} === Sine by Cosine === {{:Simpson's Formulas/Sine by Cosine}} === Cosine by Sine === {{:Simpson's Formulas/Cosine by Sine}}
Prosthaphaeresis Formulas
=== Sine plus Sine === {{:Prosthaphaeresis Formulas/Sine plus Sine}} === Sine minus Sine === {{:Prosthaphaeresis Formulas/Sine minus Sine}} === Cosine plus Cosine === {{:Prosthaphaeresis Formulas/Cosine plus Cosine}} === Cosine minus Cosine === {{:Prosthaphaeresis Formulas/Cosine minus Cosine}}
Sum of Tangent and Cotangent
:$\tan x + \cot x = \sec x \csc x$
Tangent times Tangent plus Cotangent
:$\tan x \paren {\tan x + \cot x} = \sec^2 x$
Minor Trigonometric Identities
=== Sum of Tangent and Cotangent === {{:Sum of Tangent and Cotangent}} === Tangent times Tangent plus Cotangent === {{:Tangent times Tangent plus Cotangent}} === Secant Minus Cosine === {{:Secant Minus Cosine}} === Square of Tangent Minus Square of Sine === {{:Square of Tangent Minus Square of Sine}} === Difference of Fourth Powers of Cosine and Sine === {{:Difference of Fourth Powers of Cosine and Sine}} === Cosecant Minus Sine === {{:Cosecant Minus Sine}} === Cotangent Minus Tangent === {{:Cotangent Minus Tangent}} === Sum of Cosecant and Cotangent === {{:Sum of Cosecant and Cotangent}} === Sum of Squares of Secant and Cosecant === {{:Sum of Squares of Secant and Cosecant}} === Difference of Fourth Powers of Secant and Tangent === {{:Difference of Fourth Powers of Secant and Tangent}} === Reciprocal of One Plus Sine === {{:Reciprocal of One Plus Sine}} === Reciprocal of One Minus Sine === {{:Reciprocal of One Minus Sine}} === Sum of Reciprocals of One Plus and Minus Sine === {{:Sum of Reciprocals of One Plus and Minus Sine}} === Difference of Reciprocals of One Plus and Minus Sine === {{:Difference  of Reciprocals of One Plus and Minus Sine}} === Reciprocal of One Plus Cosine === {{:Reciprocal of One Plus Cosine}} === Reciprocal of One Minus Cosine === {{:Reciprocal of One Minus Cosine}} === Sum of Secant and Tangent === {{:Sum of Secant and Tangent}} === Cosine over Sum of Secant and Tangent === {{:Cosine over Sum of Secant and Tangent}} === Secant Plus One over Secant Squared === {{:Secant Plus One over Secant Squared}} === Sine Plus Cosine times Tangent Plus Cotangent === {{:Sine Plus Cosine times Tangent Plus Cotangent}} === Tangent over Secant Plus One === {{:Tangent over Secant Plus One}} === Squares of Linear Combination of Sine and Cosine === {{:Squares of Linear Combination of Sine and Cosine}} === Reciprocal of One Minus Secant === {{:Reciprocal of One Minus Secant}} === Reciprocal of One Plus Cosecant === {{:Reciprocal of One Plus Cosecant}}
Secant Minus Cosine
:$\sec x - \cos x = \sin x \tan x$
Square of Tangent Minus Square of Sine
:$\tan^2 x - \sin^2 x = \tan^2 x \ \sin^2 x$
Difference of Fourth Powers of Cosine and Sine
:$\sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x$
Cosecant Minus Sine
:$\csc x - \sin x = \cos x \ \cot x$
Sum of Squares of Secant and Cosecant
:$\sec^2 x + \csc^2 x = \sec^2 x \csc^2 x$
Difference of Fourth Powers of Secant and Tangent
:$\sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x$
Sum of Reciprocals of One Plus and Minus Sine
:$\dfrac 1 {1 - \sin x} + \dfrac 1 {1 + \sin x} = 2 \sec^2 x$
Difference of Reciprocals of One Plus and Minus Sine
:$\displaystyle \frac 1 {1 - \sin x} - \frac 1 {1 + \sin x} = 2 \tan x \ \sec x$
Sum of Secant and Tangent
:$\sec x + \tan x = \dfrac {1 + \sin x} {\cos x}$
Cosine over Sum of Secant and Tangent
:$\dfrac {\cos x} {\sec x + \tan x} = 1 - \sin x$
Secant Plus One over Secant Squared
:$\displaystyle \frac {\sec x + 1} {\sec^2 x} = \frac {\sin^2 x} {\sec x - 1}$
Sine Plus Cosine times Tangent Plus Cotangent
:$\paren {\sin x + \cos x} \paren {\tan x + \cot x} = \sec x + \csc x$
Tangent over Secant Plus One
:$\displaystyle \frac {\tan x} {\sec x + 1} = \frac {\sec x - 1} {\tan x}$
Squares of Linear Combination of Sine and Cosine
:$\left({a \cos x + b \sin x}\right)^2 + \left({b \cos x - a \sin x}\right)^2 = a^2 + b^2$
Reciprocal of One Minus Secant
:$\dfrac {\sin^2 x + 2 \cos x - 1} {\sin^2 x + 3 \cos x - 3} = \dfrac 1 {1 - \sec x}$
Reciprocal of One Plus Cosecant
:$\dfrac {\cos^2 x + 3 \sin x - 1} {\cos^2 x + 2 \sin x + 2} = \dfrac 1 {1 + \csc x}$
Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE/General Result
Let $n \in \Z_{>0}$ be a strictly positive integer. The ordinary differential equation: :$a_n x^n \, \map {f^{\paren n} } x + \dotsb + a_1 x \, \map {f'} c + a_0 \, \map f x = 0$ can be transformed to a linear differential equation with constant coefficients by substitution $x = e^t$.
Condition for Composite Relation with Inverse to be Identity
Let $\RR \subseteq S \times T$ be a relation on $S \times T$. Let $\RR \circ \RR^{-1}$ be the composite of $\RR$ with its inverse. Let $I_T$ be the identity mapping on $T$. Then: :$\RR \circ \RR^{-1} = I_T$ : :$\RR$ is many-to-one and: :$\RR$ is right-total.
Set of All Relations is a Monoid
The set of all relations $\Bbb E = \left\{{\mathcal R: \mathcal R \subseteq S \times S}\right\}$ on a set $S$ forms a monoid in which the operation is composition of relations. The only invertible elements of $\Bbb E$ are permutations. If $\mathcal R$ is invertible, its inverse is $\mathcal R^{-1}$.
Inverses of Right-Total and Left-Total Relations
Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse of $\mathcal R$. Then: