avewright/proofwiki-math huggingface dataset:
Theorem: Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine

:$\cosh x + \cosh y = 2 \map \cosh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$

Proof:
{{begin-eqn}}
{{eqn | o = 
      | r = 2 \map \cosh {\frac {x + y} 2} \map \cosh {\frac {x - y} 2}
      | c = 
}}
{{eqn | r = 2 \frac {\map \cosh {\dfrac {x + y} 2 + \dfrac {x - y} 2} + \map \cosh {\dfrac {x + y} 2 - \dfrac {x - y} 2} } 2
      | c = Werner Formula for Hyperbolic Cosine by Hyperbolic Cosine
}}
{{eqn | r = \cosh \frac {2 x} 2 + \cosh \frac {2 y} 2
      | c = 
}}
{{eqn | r = \cosh x + \cosh y
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Werner Formulas/Hyperbolic Sine by Hyperbolic Sine

:$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$

Proof:
{{begin-eqn}}
{{eqn | o = 
      | r = \frac {\map \cosh {x + y} - \map \cosh {x - y} } 2
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} - \map \cosh {x - y} } 2
      | c = Hyperbolic Cosine of Sum
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} - \paren {\cosh x \cosh y - \sinh x \sinh y} } 2
      | c = Hyperbolic Cosine of Difference
}}
{{eqn | r = \frac {2 \sinh x \sinh y} 2
}}
{{eqn | r = \sinh x \sinh y
}}
{{end-eqn}}
{{qed}}
Theorem: Werner Formulas/Hyperbolic Sine by Hyperbolic Sine

:$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sinh x \sinh y
      | r = \frac {e^x - e^{-x} } 2 \frac {e^y - e^{-y} } 2
      | c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{x + y} - e^{x - y} - e^{-x + y} + e^{-x - y} } 4
      | c = simplifying
}}
{{eqn | r = \frac 1 2 \paren {\dfrac {e^{x + y} + e^{-\paren {x + y} } } 2 - \frac {e^{x - y} + e^{-\paren {x - y} } } 2}
      | c = rearranging
}}
{{eqn | r = \frac {\cosh \paren {x + y} - \cosh \paren {x - y} } 2
      | c = {{Defof|Hyperbolic Cosine}}
}}
{{end-eqn}}
{{qed}}
Theorem: Werner Formulas/Hyperbolic Sine by Hyperbolic Sine

:$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sinh x \sinh y
      | r = i^2 \map \sin {\frac x i} \map \sin {\frac y i}
      | c = Sine in terms of Hyperbolic Sine
}}
{{eqn | r = -\map \sin {\frac x i} \map \sin {\frac y i}
      | c = $i^2 = -1$
}}
{{eqn | r = -\frac {\map \cos {\frac x i - \frac y i} - \map \cos {\frac x i + \frac y i} } 2
      | c = Werner Formula for Sine by Sine
}}
{{eqn | r = \frac {\map \cos {\frac x i + \frac y i} - \map \cos {\frac x i - \frac y i} } 2
      | c = simplifying
}}
{{eqn | r = \frac {\map \cosh {x + y} - \map \cosh {x - y} } 2
      | c = Cosine in terms of Hyperbolic Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Werner Formulas/Hyperbolic Cosine by Hyperbolic Cosine

:$\cosh x \cosh y = \dfrac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$

Proof:
{{begin-eqn}}
{{eqn | o = 
      | r = \frac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \cosh \paren {x - y} } 2
      | c = Hyperbolic Cosine of Sum
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \paren {\cosh x \cosh y - \sinh x \sinh y} } 2
      | c = Hyperbolic Cosine of Difference
}}
{{eqn | r = \frac {2 \cosh x \cosh y} 2
}}
{{eqn | r = \cosh x \cosh y
}}
{{end-eqn}}
{{qed}}
Theorem: Werner Formulas/Hyperbolic Sine by Hyperbolic Cosine

:$\sinh x \cosh y = \dfrac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2$

Proof:
{{begin-eqn}}
{{eqn | o = 
      | r = \frac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2
}}
{{eqn | r = \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \cosh \paren {x - y} } 2
      | c = Hyperbolic Sine of Sum
}}
{{eqn | r = \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \paren {\sinh x \cosh y - \cosh x \sinh y} } 2
      | c = Hyperbolic Sine of Difference
}}
{{eqn | r = \frac {2 \sinh x \cosh y} 2
}}
{{eqn | r = \sinh x \cosh y
}}
{{end-eqn}}
{{qed}}
Theorem: Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant

Everywhere that the function is defined:
:$\map \arsinh {\dfrac 1 x} = \arcsch x$
where $\arsinh$ and $\arcsch$ denote real area hyperbolic sine and real area hyperbolic cosecant respectively.

Proof:
{{begin-eqn}}
{{eqn | l = \map \arsinh {\dfrac 1 x}
      | r = y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = \frac 1 x
      | r = \sinh y
      | c = {{Defof|Real Area Hyperbolic Sine}}
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = \csch y
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | ll= \leadstoandfrom
      | l = \arcsch x
      | r = y
      | c = {{Defof|Real Area Hyperbolic Cosecant}}
}}
{{end-eqn}}
{{qed}}
Theorem: Real Area Hyperbolic Cosine of Reciprocal equals Real Area Hyperbolic Secant

Everywhere that the function is defined:
:$\map \arcosh {\dfrac 1 x} = \arsech x$
where $\arcosh$ and $\arsech$ denote real area hyperbolic cosine and real area hyperbolic secant respectively.

Proof:
{{begin-eqn}}
{{eqn | l = \map \arcosh {\dfrac 1 x}
      | r = y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = \frac 1 x
      | r = \cosh y
      | c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = \sech y
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | ll= \leadstoandfrom
      | l = \arsech x
      | r = y
      | c = {{Defof|Real Area Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}}
Theorem: Real Area Hyperbolic Tangent of Reciprocal equals Real Area Hyperbolic Cotangent

Everywhere that the function is defined:
:$\map \artanh {\dfrac 1 x} = \arcoth x$
where $\artanh$ and $\arcoth$ denote real area hyperbolic tangent and real area hyperbolic cotangent respectively.

Proof:
{{begin-eqn}}
{{eqn | l = \map \artanh {\dfrac 1 x}
      | r = y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = \frac 1 x
      | r = \tanh y
      | c = {{Defof|Real Area Hyperbolic Tangent}}
}}
{{eqn | ll= \leadstoandfrom
      | l = \frac 1 x
      | r = \frac {\cosh y} {\sinh y}
      | c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = \frac {\sinh y} {\cosh y}
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = \coth y
      | c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | ll= \leadstoandfrom
      | l = \arcoth x
      | r = y
      | c = {{Defof|Real Area Hyperbolic Cotangent}}
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Hyperbolic Sine is Odd Function

Let $x \in \R$.
Then:
:$\map \arsinh {-x} = -\arsinh x$
where $\arsinh$ denotes the inverse hyperbolic sine function.

Proof:
{{begin-eqn}}
{{eqn | l = \map \arsinh {-x}
      | r = y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = -x
      | r = \sinh y
      | c = {{Defof|Inverse Hyperbolic Sine|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = -\sinh y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = \map \sinh {-y}
      | c = Hyperbolic Sine Function is Odd
}}
{{eqn | ll= \leadstoandfrom
      | l = \arsinh x
      | r = -y
      | c = {{Defof|Inverse Hyperbolic Sine|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
      | l = -\arsinh x
      | r = y
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Hyperbolic Tangent is Odd Function

:$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\tanh^{-1} } {-x}
      | r = y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = -x
      | r = \tanh y
      | c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = -\tanh y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = \map \tanh {-y}
      | c = Hyperbolic Tangent Function is Odd
}}
{{eqn | ll= \leadstoandfrom
      | l = \tanh^{-1} x
      | r = -y
      | c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 1}}
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Hyperbolic Tangent is Odd Function

:$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$

Proof:
{{begin-eqn}}
{{eqn	| l = \map {\tanh^{-1} } {-x}
	| r = \frac 1 2 \map \ln {\frac {1 + \paren {-x} } {1 - \paren {-x} } }
	| c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 2}}
}}
{{eqn	| r = \frac 1 2 \map \ln {\frac {1 - x} {1 + x} }
}}
{{eqn	| r = \frac 1 2 \paren {\map \ln {1 - x} - \map \ln {1 + x} }
	| c = Difference of Logarithms
}}
{{eqn	| r = -\frac 1 2 \paren {\map \ln {1 + x} - \map \ln {1 - x} }
	| c = 
}}
{{eqn	| r = -\frac 1 2 \map \ln {\frac {1 + x} {1 - x} }
	| c = Difference of Logarithms
}}
{{eqn	| r = -\tanh^{-1} x
	| c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 2}}
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Hyperbolic Cotangent is Odd Function

:$\map {\coth^{-1} } {-x} = -\coth^{-1} x$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\coth^{-1} } {-x}
      | r = y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = -x
      | r = \coth y
      | c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = -\coth y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = \map \coth {-y}
      | c = Hyperbolic Cotangent Function is Odd
}}
{{eqn | ll= \leadstoandfrom
      | l = \coth^{-1} x
      | r = -y
      | c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 1}}
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Hyperbolic Cotangent is Odd Function

:$\map {\coth^{-1} } {-x} = -\coth^{-1} x$

Proof:
{{begin-eqn}}
{{eqn	| l = \map {\coth^{-1} } {-x}
	| r = \frac 1 2 \map \ln {\frac {-z + 1} {-z - 1} }
	| c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 2}}
}}
{{eqn	| r = \frac 1 2 \map \ln {\frac {z - 1} {z + 1} }
	| c = multiplying the argument by $\dfrac {-1} {-1}$
}}
{{eqn	| r = \frac 1 2 \paren {\map \ln {z - 1} - \map \ln {z + 1} }
	| c = Difference of Logarithms
}}
{{eqn	| r = -\frac 1 2 \paren {\map \ln {z + 1} - \map \ln {z - 1} }
	| c = 
}}
{{eqn	| r = -\frac 1 2 \map \ln {\frac {z + 1} {z - 1} }
	| c = Difference of Logarithms
}}
{{eqn	| r = -\coth^{-1} x
	| c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 2}}
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Hyperbolic Cosecant is Odd Function

Let $x \in \R$.
Then:
:$\map {\csch^{-1} } {-x} = -\csch^{-1} x$
where $\map {\csch^{-1} } {-x}$ denotes the inverse hyperbolic cosecant function.

Proof:
{{begin-eqn}}
{{eqn | l = \map {\csch^{-1} } {-x}
      | r = y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = -x
      | r = \csch y
      | c = {{Defof|Inverse Hyperbolic Cosecant|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = -\csch y
      | c = 
}}
{{eqn | ll= \leadstoandfrom
      | l = x
      | r = \map \csch {-y}
      | c = Hyperbolic Cosecant Function is Odd
}}
{{eqn | ll= \leadstoandfrom
      | l = \csch ^{-1} x
      | r = -y
      | c = {{Defof|Inverse Hyperbolic Cosecant|subdef = Real|index = 1}}
}}
{{end-eqn}}
{{qed}}
Theorem: Hyperbolic Cosecant in terms of Cosecant

Let $z \in \C$ be a complex number.
Then:
:$i \csch z = -\csc \paren {i z}$
where:
: $\csc$ denotes the cosecant function
: $\csch$ denotes the hyperbolic cosecant
: $i$ is the imaginary unit: $i^2 = -1$.

Proof:
{{begin-eqn}} 
{{eqn | l = i \csch  z
      | r = \frac i {\sinh z}
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {-1} {i \sinh z}
      | c = as $i^2 = -1$
}}
{{eqn | r = \frac {-1} {\sin \paren {i z} }
      | c = Hyperbolic Sine in terms of Sine
}}
{{eqn | r = -\csc \paren {i z}
      | c = {{Defof|Complex Cosecant Function}}
}}
{{end-eqn}}
{{qed}}
Theorem: Hyperbolic Secant in terms of Secant

Let $z \in \C$ be a complex number.
Then:
:$\sech z = \sec \paren {i z}$
where:
: $\sec$ denotes the secant function
: $\sech$ denotes the hyperbolic secant
: $i$ is the imaginary unit: $i^2 = -1$.

Proof:
{{begin-eqn}} 
{{eqn | l = \sec \paren {i z}
      | r = \frac 1 {\cos \paren {i z} }
      | c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \frac 1 {\cosh z}
      | c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \sech z
      | c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}}
Theorem: Hyperbolic Cotangent in terms of Cotangent

Let $z \in \C$ be a complex number.
Then:
:$\coth z = -\cot \paren {i z}$
where:
: $\cot$ denotes the cotangent function
: $\coth$ denotes the hyperbolic cotangent
: $i$ is the imaginary unit: $i^2 = -1$.

Proof:
{{begin-eqn}} 
{{eqn | l = i \coth z
      | r = \frac {i \cosh z} {\sinh z}
      | c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {-\cosh z} {i \sinh z}
      | c = $i^2 = -1$
}}
{{eqn | r = \frac {-\cos \paren {i z} } {i \sinh z}
      | c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \frac {-\cos \paren {i z} } {\sin \paren {i z} }
      | c = Hyperbolic Sine in terms of Sine
}}
{{eqn | r = -\cot \paren {i z}
      | c = {{Defof|Complex Cotangent Function}}
}}
{{end-eqn}}
{{qed}}
Theorem: Sine in terms of Hyperbolic Sine

Let $z \in \C$ be a complex number.
Then:
:$i \sin z = \map \sinh {i z}$
where:
:$\sin$ denotes the complex sine
:$\sinh$ denotes the hyperbolic sine
:$i$ is the imaginary unit: $i^2 = -1$.

Proof:
{{begin-eqn}} 
{{eqn | l = \map \sinh {i z}
      | r = \frac {e^{i z} - e^{-i z} } 2
      | c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = i \frac {e^{i z} - e^{-i z} } {2 i}
      | c = multiplying top and bottom by $i$
}}
{{eqn | r = i \sin z
      | c = Euler's Sine Identity
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine in terms of Hyperbolic Cosine

Let $z \in \C$ be a complex number.
Then:
:$\cos z = \map \cosh {i z}$
where:
:$\cos$ denotes the complex cosine
:$\cosh$ denotes the hyperbolic cosine
:$i$ is the imaginary unit: $i^2 = -1$.

Proof:
{{begin-eqn}} 
{{eqn | l = \map \cosh {i z}
      | r = \frac {e^{i z} + e^{-i z} } 2
      | c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \cos z
      | c = Euler's Cosine Identity
}}
{{end-eqn}}
{{qed}}
Theorem: Tangent in terms of Hyperbolic Tangent

Let $z \in \C$ be a complex number.
Then:
:$i \tan z = \map \tanh {i z}$
where:
:$\tan$ denotes the tangent function
:$\tanh$ denotes the hyperbolic tangent
:$i$ is the imaginary unit: $i^2 = -1$.

Proof:
{{begin-eqn}} 
{{eqn | l = \map \tanh {i z}
      | r = \frac {\map \sinh {i z} } {\map \cosh {i z} }
      | c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac {i \sin z} {\map \cosh {i z} }
      | c = Sine in terms of Hyperbolic Sine
}}
{{eqn | r = \frac {i \sin z} {\cos z}
      | c = Cosine in terms of Hyperbolic Cosine
}}
{{eqn | r = i \tan z
      | c = {{Defof|Complex Tangent Function}}
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant in terms of Hyperbolic Cosecant

Let $z \in \C$ be a complex number.
Then:
:$i \csc = -\csch \paren {i z}$
where:
: $\csc$ denotes the cosecant function
: $\csch$ denotes the hyperbolic cosecant
: $i$ is the imaginary unit: $i^2 = -1$.

Proof:
{{begin-eqn}} 
{{eqn | l = i \csc x
      | r = \frac i {\sin z}
      | c = {{Defof|Complex Cosecant Function}}
}}
{{eqn | r = \frac 1 {-i \sin z}
      | c = $i^2 = -1$
}}
{{eqn | r = \frac 1 {-\sinh \paren {i z} }
      | c = Sine in terms of Hyperbolic Sine
}}
{{eqn | r = -\csch \paren {i z}
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{end-eqn}}
{{qed}}
Theorem: Secant in terms of Hyperbolic Secant

Let $z \in \C$ be a complex number.
Then:
:$\sec z = \map \sech {i z}$
where:
:$\sec$ denotes the secant function
:$\sech$ denotes the  hyperbolic secant
:$i$ is the imaginary unit: $i^2 = -1$.

Proof:
{{begin-eqn}} 
{{eqn | l = \sec z
      | r = \frac 1 {\cos z}
      | c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \frac 1 {\map \cosh {i z} }
      | c = Cosine in terms of Hyperbolic Cosine
}}
{{eqn | r = \map \sech {i z}
      | c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent in terms of Hyperbolic Cotangent

Let $z \in \C$ be a complex number.
Then:
:$i \cot z = -\map \coth {i z}$
where:
:$\cot$ denotes the cotangent function
:$\coth$ denotes the hyperbolic cotangent
:$i$ is the imaginary unit: $i^2 = -1$.

Proof:
{{begin-eqn}} 
{{eqn | l = i \cot z
      | r = i \frac {\cos z} {\sin z}
      | c = {{Defof|Complex Cotangent Function}}
}}
{{eqn | r = -\frac {\cos z} {i \sin z}
      | c = $i^2 = -1$
}}
{{eqn | r = -\frac {\map \cosh {i z} } {i \sin z}
      | c = Cosine in terms of Hyperbolic Cosine
}}
{{eqn | r = -\frac {\map \cosh {i z} } {\map \sinh {i z} }
      | c = Sine in terms of Hyperbolic Sine
}}
{{eqn | r = -\map \coth {i z}
      | c = {{Defof|Hyperbolic Cotangent}}
}}
{{end-eqn}}
{{qed}}
Theorem: General Periodicity Property

Let $f: \R \to \R$ be a periodic real function.
Let $L$ be a periodic element of $f$.
Then:
:$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$
That is, after every distance $L$, the function $f$ repeats itself.

Proof:
There are two cases to consider: either $n$ is not negative, or it is negative.
Since the Natural Numbers are Non-Negative Integers, the case where $n \ge 0$ will be proved using induction.
Theorem: Periodicity of Hyperbolic Sine

Let $k \in \Z$.
Then:
:$\map \sinh {x + 2 k \pi i} = \sinh x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \sinh {x + 2 k \pi i}
      | r = \frac {e^{x + 2 k \pi i} - e^{-\paren {x + 2 k \pi i} } } {2 i}
      | c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{i \paren {-i x + 2 k \pi} } - e^{i \paren {i x + 2 \paren {-k} \pi} } } {2 i}
      | c = $i^2 = -1$ and simplifying
}}
{{eqn | r = \frac {e^{i \paren {-i x} } - e^{i \paren {i x} } } {2 i}
      | c = Period of Complex Exponential Function
}}
{{eqn | r = \frac {e^x - e^{- x} } {2 i}
      | c = $i^2 = -1$
}}
{{eqn | r = \sinh x
      | c = {{Defof|Hyperbolic Sine}}
}}
{{end-eqn}}
{{qed}}
Theorem: Periodicity of Hyperbolic Cosine

Let $k \in \Z$.
Then:
:$\map \cosh {x + 2 k \pi i} = \cosh x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \cosh {x + 2 k \pi i}
      | r = \frac {e^{x + 2 k \pi i} + e^{- \paren {x + 2 k \pi i} } } {2 i}
      | c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{i \paren {-i x + 2 k \pi} } + e^{i \paren {i x + 2 \paren {-k} \pi} } } {2 i}
      | c = $i^2 = -1$ and simplifying
}}
{{eqn | r = \frac {e^{i \paren {-i x} } + e^{i \paren {i x} } } {2 i}
      | c = Period of Complex Exponential Function
}}
{{eqn | r = \frac {e^x + e^{- x} } {2 i}
      | c = $i^2 = -1$
}}
{{eqn | r = \cosh x
      | c = {{Defof|Hyperbolic Cosine}}
}}
{{end-eqn}}
{{qed}}
Theorem: Periodicity of Hyperbolic Tangent

Let $k \in \Z$.
Then:
:$\map \tanh {x + 2 k \pi i} = \tanh x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \tanh {x + 2 k \pi i}
      | r = \frac {\map \sinh {x + 2 k \pi i} } {\map \cosh {x + 2 k \pi i} }
      | c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac {\sinh x} {\map \cosh {x + 2 k \pi i} }
      | c = Periodicity of Hyperbolic Sine
}}
{{eqn | r = \frac {\sinh x} {\cosh x}
      | c = Periodicity of Hyperbolic Cosine
}}
{{eqn | r = \tanh x
      | c = {{Defof|Hyperbolic Tangent}}
}}
{{end-eqn}}
{{qed}}
Theorem: Periodicity of Hyperbolic Cosecant

Let $k \in \Z$.
Then:
:$\map \csch {x + 2 k \pi i} = \csch x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \csch {x + 2 k \pi i}
      | r = \frac 1 {\map \sinh {x + 2 k \pi i} }
      | c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \frac 1 {\sinh x}
      | c = Periodicity of Hyperbolic Sine
}}
{{eqn | r = \csch x
      | c = {{Defof|Hyperbolic Cosecant}}
}}
{{end-eqn}}
{{qed}}
Theorem: Periodicity of Hyperbolic Secant

Let $k \in \Z$.
Then:
:$\map \sech {x + 2 k \pi i} = \sech x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \sech {x + 2 k \pi i}
      | r = \frac 1 {\map \cosh {x + 2 k \pi i} }
      | c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 1 {\cosh x}
      | c = Periodicity of Hyperbolic Cosine
}}
{{eqn | r = \sech
      | c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}}
Theorem: Periodicity of Hyperbolic Cotangent

Let $k \in \Z$.
Then:
:$\map \coth {x + 2 k \pi i} = \coth x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \coth {x + 2 k \pi i}
      | r = \frac {\map \cosh {x + 2 k \pi i} } {\map \sinh {x + 2 k \pi i} }
      | c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\map \cosh {x + 2 k \pi i} } {\sinh x}
      | c = Periodicity of Hyperbolic Sine
}}
{{eqn | r = \frac {\cosh x} {\sinh x}
      | c = Periodicity of Hyperbolic Cosine
}}
{{eqn | r = \coth x
      | c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Sine of Imaginary Number

:$\map {\sin^{-1} } {i x} = i \sinh^{-1} x$

Proof:
{{begin-eqn}}
{{eqn | l = y
      | r = \map {\sin^{-1} } {i x}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \sin y
      | r = i x
      | c = {{Defof|Complex Inverse Sine}}
}}
{{eqn | ll= \leadsto
      | l = i \sin y
      | r = -x
      | c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
      | l = \map {\sin^{-1} } {i y}
      | r = -x
      | c = Sine in terms of Hyperbolic Sine
}}
{{eqn | ll= \leadsto
      | l = i y
      | r = \map {\sinh^{-1} } {-x}
      | c = {{Defof|Inverse Hyperbolic Sine}}
}}
{{eqn | ll= \leadsto
      | l = i y
      | r = -\sinh^{-1} x
      | c = Inverse Hyperbolic Sine is Odd Function
}}
{{eqn | ll= \leadsto
      | l = y
      | r = i \sinh^{-1} x
      | c = multiplying both sides by $-i$
}}
{{end-eqn}}
{{qed}}
{{explain|Investigate the multifunctional nature of Definition:Complex Inverse Sine and similarly for Hyperbolic Sine}}
Theorem: Inverse Hyperbolic Sine of Imaginary Number

:$\map {\sinh^{-1} } {i x} = i \sin^{-1} x$

Proof:
{{begin-eqn}}
{{eqn | l = y
      | r = \map {\sinh^{-1} } {i x} 
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \sinh y
      | r = i x
      | c = {{Defof|Inverse Hyperbolic Sine}}
}}
{{eqn | ll= \leadsto
      | l = i \sinh y
      | r = - x
      | c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
      | l = \map \sin {i y}
      | r = -x
      | c = Hyperbolic Sine in terms of Sine
}}
{{eqn | ll= \leadsto
      | l = i y
      | r = \map {\sin^{-1} } {-x}
      | c = {{Defof|Complex Inverse Sine}}
}}
{{eqn | ll= \leadsto
      | l = i y
      | r = -\sin^{-1} x
      | c = Inverse Sine is Odd Function
}}
{{eqn | ll= \leadsto
      | l = y
      | r = i \sinh^{-1} x
      | c = multiplying both sides by $-i$
}}
{{end-eqn}}
{{qed}}
{{explain|Investigate the multifunctional nature of Definition:Complex Inverse Sine and similarly for Hyperbolic Sine}}
Theorem: Inverse Cosine of Imaginary Number

:$\cos^{-1} x = \pm \, i \cosh^{-1} x$

Proof:
{{begin-eqn}}
{{eqn | l = y
      | r = \cos^{-1} x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \cos y
      | r = x
      | c = {{Defof|Inverse Cosine}}
}}
{{eqn | ll= \leadsto
      | l = \map \cos {\pm \, y}
      | r = x
      | c = Cosine Function is Even
}}
{{eqn | ll= \leadsto
      | l = \map \cosh {\pm \, i y}
      | r = x
      | c = Cosine in terms of Hyperbolic Cosine
}}
{{eqn | ll= \leadsto
      | l = \pm \, i y
      | r = \cosh^{-1} x
      | c = {{Defof|Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
      | l = y
      | r = \pm \, i \cosh^{-1} x
      | c = multiplying both sides by $\pm \, i$
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Hyperbolic Cosine of Imaginary Number

:$\cosh^{-1} x = \pm \, i \cos^{-1} x$

Proof:
{{begin-eqn}}
{{eqn | l = y
      | r = \cosh^{-1} x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \cosh y
      | r = x
      | c = {{Defof|Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
      | l = \map \cosh {\pm \, y}
      | r = x
      | c = Hyperbolic Cosine Function is Even
}}
{{eqn | ll= \leadsto
      | l = \map \cos {\pm \, i y}
      | r = x
      | c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | ll= \leadsto
      | l = \pm \, i y
      | r = \cos^{-1} x
      | c = {{Defof|Inverse Cosine}}
}}
{{eqn | ll= \leadsto
      | l = y
      | r = \pm \, i \cos^{-1} x
      | c = multiplying both sides by $\pm \, i$
}}
{{end-eqn}}
{{qed}}