hcju/proofwikiqa huggingface dataset:
{{begin-eqn}} {{eqn | l = \tanh \frac x 2       | r = \pm \sqrt {\frac {\cosh x - 1} {\cosh x + 1} }       | c = Half Angle Formula for Hyperbolic Tangent }} {{eqn | r = \pm \sqrt {\frac {\paren {\cosh x - 1}^2} {\paren {\cosh x + 1} \paren {\cosh x - 1} } }       | c = multiplying numerator and denominator by $\sqrt {\cosh x - 1}$ }} {{eqn | r = \pm \sqrt {\frac {\paren {\cosh x - 1}^2} {\cosh^2 x - 1} }       | c = Difference of Two Squares }} {{eqn | r = \pm \sqrt {\frac {\paren {\cosh x - 1}^2} {\sinh^2 x} }       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = \pm \frac {\cosh x - 1} {\sinh x} }} {{end-eqn}} Since $\cosh x \ge 1$, it follows that $\cosh x - 1 \ge 0$, with equality happening at $x = 0$. We also have that: :when $x > 0$, $\tanh \dfrac x 2 > 0$ and $\sinh x > 0$ :when $x < 0$, $\tanh \dfrac x 2 < 0$ and $\sinh x < 0$. Thus: :$\tanh \dfrac x 2 = \dfrac {\cosh x - 1} {\sinh x}$ {{qed}}
{{begin-eqn}} {{eqn | l = \sinh {3 x}       | r = \map \sinh {2 x + x} }} {{eqn | r = \sinh 2 x \cosh x + \cosh 2 x \sinh x       | c = Hyperbolic Sine of Sum }} {{eqn | r = \paren {2 \sinh x \cosh x} \cosh x + \cosh 2 x \sinh x       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \paren {2 \sinh x \cosh x} \cosh x + \paren {\cosh^2 x + \sinh^2 x} \sinh x       | c = Double Angle Formula for Hyperbolic Cosine }} {{eqn | r = 2 \sinh x \cosh^2 x + \cosh^2 x \sinh x + \sinh^3 x       | c = multiplying out }} {{eqn | r = 2 \sinh x \paren {1 + \sinh^2 x} + \paren {1 + \sinh^2 x} \sinh x + \sinh^3 x       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = 2 \sinh x + 2 \sinh^3 x + \sinh x + \sinh^3 x + \sinh^3 x       | c = multiplying out }} {{eqn | r = 3 \sinh x + 4 \sinh^3 x       | c = gathering terms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cosh 3 x       | r = \cosh \paren {2 x + x} }} {{eqn | r = \cosh 2 x \cosh x + \sinh 2 x \sinh x       | c = Hyperbolic Cosine of Sum }} {{eqn | r = \paren {\cosh^2 x + \sinh^2 x} \cosh x + \sinh 2 x \sinh x       | c = Double Angle Formula for Hyperbolic Cosine }} {{eqn | r = \paren {\cosh^2 x + \sinh^2 x} \cosh x + \paren {2 \sinh x \cosh x} \sinh x       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \cosh^3 x + \sinh^2 x \cosh x + 2 \sinh^2 x \cosh x       | c = multiplying out }} {{eqn | r = \cosh^3 x + \paren {\cosh^2 x - 1} \cosh x + 2 \paren {\cosh^2 x - 1} \cosh x       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = \cosh^3 x + \cosh^3 x - \cosh x + 2 \cosh^3 x - 2 \cosh x       | c = multiplying out }} {{eqn | r = 4 \cosh^3 x - 3 \cosh x       | c = gathering terms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn|l = \tanh \left({3 x}\right)      |r = \frac {\sinh \left({3 x}\right)} {\cosh \left({3 x}\right)}      |c = Definition of Hyperbolic Tangent }} {{eqn|r = \frac {3 \sinh x + 4 \sinh^3 x} {\cosh \left({3 x}\right)}      |c = Triple Angle Formula for Hyperbolic Sine }} {{eqn|r = \frac {3 \sinh x + 4 \sinh^3 x} {4 \cosh^3 x - 3 \cosh x}      |c = Triple Angle Formula for Hyperbolic Cosine }} {{eqn|r = \frac {3 \sinh x + 4 \sinh^3 x} {4 \cosh^3 x - 3 \cosh x} \frac {\left({1 / \cosh^3 x}\right)} {\left({1 / \cosh^3 x}\right)}      |c =  }} {{eqn|r = \frac {\frac {3 \tanh x}{\cosh^2 x} + 4 \tanh^3 x} {4 - \frac {3 \cosh x}{\cosh^3 x} }      |c = Definition of Hyperbolic Tangent }} {{eqn|r = \frac {3 \tanh x \operatorname{sech}^2 x + 4 \tanh^3 x} {4 - 3 \operatorname{sech}^2 x}      |c = Definition of Hyperbolic Secant }} {{eqn|r = \frac {3 \tanh x \left({1 - \tanh^2 x}\right) + 4 \tanh^3 x} {4 - 3  \left({1 - \tanh^2 x}\right)}      |c = Sum of Squares of Hyperbolic Secant and Tangent }} {{eqn|r = \frac {3 \tanh x - 3 \tanh^3 x + 4 \tanh^3 x} {4 - 3 + 3 \tanh^2 x}      |c = multipying out }} {{eqn|r = \frac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}      |c = gathering terms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn|l = \sinh \left({4 x}\right)      |r = \sinh \left({3 x + x}\right) }} {{eqn|r = \sinh 3 x \cosh x + \cosh 3 x \sinh x      |c = Hyperbolic Sine of Sum }} {{eqn|r = \left({3 \sinh x + 4 \sinh^3 x}\right) \cosh x + \cosh 3 x \sinh x      |c = Triple Angle Formula for Hyperbolic Sine }} {{eqn|r = \left({3 \sinh x + 4 \sinh^3 x}\right) \cosh x + \left({4 \cosh^3 x - 3 \cosh x}\right) \sinh x      |c = Triple Angle Formula for Hyperbolic Cosine }} {{eqn|r = 3 \sinh x \cosh x + 4 \sinh^3 x \cosh x + 4 \cosh^3 x \sinh x - 3 \cosh x \sinh x      |c = multiplying out }} {{eqn|r = 3 \sinh x \cosh x + 4 \sinh^3 x \cosh x      |c =  }} {{eqn|o =       |ro= +      |r = 4 \cosh x \left({1 + \sinh^2 x}\right) \sinh x - 3 \cosh x \sinh x      |c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn|r = 8 \sinh^3 x \cosh x + 4 \sinh x \cosh x      |c = multiplying out and gathering terms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn|l = \cosh \left({4 x}\right)      |r = \cosh \left({2 x + 2 x}\right) }} {{eqn|r = \cosh 2 x \cosh 2 x + \sinh 2 x \sinh 2 x      |c = Hyperbolic Cosine of Sum }} {{eqn|r = \left({\cosh^2 x + \sinh^2 x}\right) \left({\cosh^2 x + \sinh^2 x}\right) + \left({2 \sinh x \cosh x}\right) \left({2 \sinh x \cosh x}\right)      |c = Double Angle Formulas }} {{eqn|r = \cosh^4 x + 2 \cosh^2 x \sinh^2 x + \sinh^4 x + 4 \cosh^2 x \sinh^2 x      |c = multiplying out }} {{eqn|r = \cosh^4 x + 2 \cosh^2 x \left({\cosh^2 x - 1}\right) + \left({\cosh^2 x - 1}\right)^2 + 4 \cosh^2 x \left({\cosh^2 x - 1}\right)      |c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn|r = 8 \cosh^4 x - 8 \cosh^2 x + 1      |c = multiplying out and gathering terms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn|l = \tanh \left({4 x}\right)      |r = \frac {\sinh \left({4 x}\right)} {\cosh \left({4 x}\right)}      |c = Definition of Hyperbolic Tangent }} {{eqn|r = \frac {8 \sinh^3 x \cosh x + 4 \sinh x \cosh x} {\cosh \left({4 x}\right)}      |c = Quadruple Angle Formula for Hyperbolic Sine }} {{eqn|r = \frac {8 \sinh^3 x \cosh x + 4 \sinh x \cosh x} {8 \cosh^4 x - 8 \cosh^2 x + 1}      |c = Quadruple Angle Formula for Hyperbolic Cosine }} {{eqn|r = \frac {8 \tanh^3 x + 4 \frac {\tanh x} {\cosh^2 x} } {8 - \frac 8 {\cosh^2 x} + \frac 1 {\cosh^4 x} }      |c = dividing top and bottom by $\cosh^4 x$ }} {{eqn|r =  \frac {8 \tanh^3 x + 4 \tanh x \operatorname{sech}^2 x} {8 - 8 \operatorname{sech}^2 x + \operatorname{sech}^4 x}      |c = Definition of Hyperbolic Secant }} {{eqn|r =  \frac {8 \tanh^3 x + 4 \tanh x \left({1 - \tanh^2 x}\right)} {8 - 8 \left({1 - \tanh^2 x}\right) + \left({1 - \tanh^2 x}\right)^2}      |c = Sum of Squares of Hyperbolic Secant and Tangent }} {{eqn|r =  \frac {4 \tanh^3 x + 4 \tanh x} {1 + 6 \tanh^2 x + \tanh^4 x}      |c = multiplying out and gathering terms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sinh 3 x       | r = 3 \sinh x + 4 \sinh^3 x       | c = Triple Angle Formula for Hyperbolic Sine }} {{eqn | ll= \leadsto       | l = 4 \sinh^3 x       | r = \sinh 3 x - 3 \sinh x       | c = rearranging }} {{eqn | ll= \leadsto       | l = \sinh^3 x       | r = \frac {\sinh 3 x - 3 \sinh x} 4       | c = dividing both sides by $4$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn|l = \sinh^4 x      |r = \left({\sinh^2 x}\right)^2 }} {{eqn|r = \left({\frac {\cosh 2 x - 1} 2}\right)^2      |c = Square of Hyperbolic Sine }} {{eqn|r = \frac {\cosh^2 2 x - 2 \cosh 2 x + 1} 4      |c = multiplying out }} {{eqn|r = \frac {\frac {\cosh 4 x + 1} 2 - 2 \cosh 2 x + 1} 4      |c = Square of Hyperbolic Cosine }} {{eqn|r = \frac {\cosh 4 x + 1 - 4 \cosh 2 x + 2} 8      |c = multiplying top and bottom by $2$ }} {{eqn|r = \frac {3 - 4 \cosh 2 x + \cosh 4 x} 8      |c = rearrangement }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | o =        | r = 2 \sinh \left({\frac {x + y} 2}\right) \cosh \left({\frac {x - y} 2}\right)       | c =  }} {{eqn | r = 2 \frac {\sinh \left({\dfrac {x + y} 2 + \dfrac {x - y} 2}\right) + \sinh \left({\dfrac {x + y} 2 - \dfrac {x - y} 2}\right)}  2       | c = Simpson's Formula for Hyperbolic Sine by Hyperbolic Cosine }} {{eqn | r = \sinh \frac {2 x} 2 + \sinh \frac {2 y} 2       | c =  }} {{eqn | r = \sinh x + \sinh y       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | o =       | r = 2 \map \cosh {\frac {x + y} 2} \map \sinh {\frac {x - y} 2}       | c =  }} {{eqn | r = 2 \frac {\map \sinh {\dfrac {x - y} 2 + \dfrac {x + y} 2} + \map \sinh {\dfrac {x - y} 2 - \dfrac {x + y} 2} } 2       | c = Simpson's Formula for Hyperbolic Sine by Hyperbolic Cosine }} {{eqn | r = \sinh \frac {2 x} 2 + \map \sinh {-\frac {2 y} 2}       | c =  }} {{eqn | r = \sinh \frac {2 x} 2 - \sinh \frac {2 y} 2       | c = Hyperbolic Sine Function is Odd }} {{eqn | r = \sinh x - \sinh y       | c =  }} {{end-eqn}} {{qed}}
{{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 = Simpson's 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}}
{{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}}
{{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}}
{{begin-eqn}} {{eqn | l = \map {\sinh^{-1} } {-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 = \sinh^{-1} x       | r = -y       | c = {{Defof|Inverse Hyperbolic Sine|subdef = Real|index = 1}} }} {{eqn | ll= \leadstoandfrom       | l = -\sinh^{-1} x       | r = y       | c =  }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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 = Sine Exponential Formulation }} {{end-eqn}} {{qed}}
{{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 = Cosine Exponential Formulation }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
{{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}}
{{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 {\cosh \paren {i z} } {i \sin z}       | c = Cosine in terms of Hyperbolic Cosine }} {{eqn | r = -\frac {\cosh \paren {i z} } {\sinh \paren {i z} }       | c = Sine in terms of Hyperbolic Sine }} {{eqn | r = -\coth \paren {i z}       | c = {{Defof|Hyperbolic Cotangent}} }} {{end-eqn}} {{qed}}
Let $X = \mathbb C$. 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. === Case 1 === ==== Basis for the Induction ==== The case for which $n = 0$ is trivial, because: :$x + 0 \cdot L = x$ ==== Induction Hypothesis ==== For some $n \in \Z_{\ge 0}$, suppose that: :$\map f x = \map f {x + n L}$ ==== Induction Step ==== For the induction step, let $n \to n + 1$. Then: {{begin-eqn}} {{eqn | l = \map f {x + \paren {n + 1} L}       | r = \map f {x + \paren {L + n L} } }} {{eqn | r = \map f {\paren {x + L} + n L}       | c = Complex Addition is Associative }} {{eqn | r = \map f {x + L}       | c = Induction Hypothesis }} {{eqn | r = \map f x       | c = Basis for the Induction }} {{end-eqn}} === Case 2 === If $n < 0$, then: {{begin-eqn}} {{eqn | l = \map f {x + n L}       | r = \map f {\paren {x + n L} - n L}       | c = Negative of Negative Number is Positive and '''Case 1''' }} {{eqn | r = \map f {x + \paren {n L - n L} }       | c = Complex Addition is Associative }} {{eqn | r = \map f x }} {{end-eqn}} Combining the results above, it is seen that for all $n \in \Z$: :$\map f x = \map f {x + n L}$ The proof for when $X = \R$ is nearly identical to the above proof. {{qed}} Category:Periodic Functions Category:Proofs by Induction o64awygd6l6v02r6a726zr6opspieft
{{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 = Periodicity 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}}
{{begin-eqn}} {{eqn | l = \cosh \left({x + 2 k \pi i}\right)       | r = \frac {e^{x + 2 k \pi i} + e^{- \left({x + 2 k \pi i}\right)} } {2 i}       | c = Definition of Hyperbolic Cosine }} {{eqn | r = \frac {e^{i \left({-i x + 2 k \pi}\right)} + e^{i \left({i x + 2 \left({-k}\right) \pi}\right)} } {2 i}       | c = $i^2 = -1$ and simplifying }} {{eqn | r = \frac {e^{i \left({-i x}\right)} + e^{i \left({i x}\right)} } {2 i}       | c = Periodicity of Complex Exponential Function }} {{eqn | r = \frac {e^x + e^{- x} } {2 i}       | c = $i^2 = -1$ }} {{eqn | r = \cosh x       | c = Definition of Hyperbolic Cosine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tanh \left({x + 2 k \pi i}\right)       | r = \frac {\sinh \left({x + 2 k \pi i}\right)} {\cosh \left({x + 2 k \pi i}\right)}       | c = Definition of Hyperbolic Tangent }} {{eqn | r = \frac {\sinh x} {\cosh \left({x + 2 k \pi i}\right)}       | c = Periodicity of Hyperbolic Sine }} {{eqn | r = \frac {\sinh x} {\cosh x}       | c = Periodicity of Hyperbolic Cosine }} {{eqn | r = \tanh x       | c = Definition of Hyperbolic Tangent }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \operatorname{csch} \left({x + 2 k \pi i}\right)       | r = \frac 1 {\sinh \left({x + 2 k \pi i}\right)}       | c = Definition of Hyperbolic Cosecant }} {{eqn | r = \frac 1 {\sinh x}       | c = Periodicity of Hyperbolic Sine }} {{eqn | r = \operatorname{csch}       | c = Definition of Hyperbolic Cosecant }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
{{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}}
{{begin-eqn}} {{eqn | l = y       | r = \sinh^{-1} \left({i x}\right)       | c =  }} {{eqn | ll= \implies       | l = \sinh y       | r = i x       | c = {{Defof|Inverse Hyperbolic Sine}} }} {{eqn | ll= \implies       | l = i \sinh y       | r = - x       | c = $i^2 = -1$ }} {{eqn | ll= \implies       | l = \sin \left({i y}\right)       | r = - x       | c = Hyperbolic Sine in terms of Sine }} {{eqn | ll= \implies       | l = i y       | r = \sin^{-1} \left({- x}\right)       | c = {{Defof|Complex Inverse Sine}} }} {{eqn | ll= \implies       | l = i y       | r = -\sin^{-1} x       | c = Inverse Sine is Odd Function }} {{eqn | ll= \implies       | 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}}
{{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}}