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Hyperbolic Functions' Identities 📂Functions

Hyperbolic Functions' Identities

Formulas

sinh⁡(−x)= −sinh⁡xcosh⁡(−x)= cosh⁡xtanh⁡(−x)= −tanh⁡xcosh⁡x+sinh⁡x= excosh⁡x−sinh⁡x= e−xcosh⁡2x−sinh⁡2x= 1 \begin{align} \sinh(-x) =&\ -\sinh x \\ \cosh(-x) =&\ \cosh x \\ \tanh(-x) =&\ - \tanh x \\ \cosh x + \sinh x =&\ e^{x} \\ \cosh x - \sinh x =&\ e^{-x} \\ \cosh^{2}x -\sinh^{2}x =&\ 1 \end{align}

Explanation

There’s really no proof needed. This can be directly known from the definition.

Proof

Proof of (1)(1)

sinh⁡(−x)= e−x−ex2=−ex−e−x2=−sinh⁡x \begin{align*} \sinh(-x) =&\ \frac{e^{-x}-e^{x}}{2} \\ =&-\frac{e^{x}-e^{-x}}{2} \\ =&-\sinh x \end{align*}

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Proof of (2)(2)

cosh⁡(−x)= e−x+ex2= ex+e−x2= cosh⁡x \begin{align*} \cosh(-x) =&\ \frac{e^{-x}+e^{x}}{2} \\ =&\ \frac{e^{x}+e^{-x}}{2} \\ =&\ \cosh x \end{align*}

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Proof of (3)(3)

tanh⁡(−x)=sinh⁡(−x)cosh⁡(−x)=−sinh⁡xcosh⁡x=−tanh⁡x \tanh (-x)=\frac{\sinh (-x)}{\cosh (-x)}=\frac{-\sinh x }{\cosh x}=-\tanh x

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Proof of (4)(4)

cosh⁡x+sinh⁡x= ex+e−x2+ex−e−x2= ex \begin{align*} \cosh x + \sinh x =&\ \frac{e^{x}+e^{-x}}{2}+\frac{e^{x}-e^{-x}}{2} \\ =&\ e^{x} \end{align*}

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Proof of (5)(5)

cosh⁡x−sinh⁡x= ex+e−x2−ex−e−x2= e−x \begin{align*} \cosh x - \sinh x =&\ \frac{e^{x}+e^{-x}}{2}-\frac{e^{x}-e^{-x}}{2} \\ =&\ e^{-x} \end{align*}

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Proof of (6)(6)

cosh⁡2x−sinh⁡2x= (ex+e−x)24−(ex−e−x)24= (e2x+e−2x+2)−(e2x+e−x−2)4= 44= 1 \begin{align*} \cosh^{2} x -\sinh^{2}x =&\ \frac{(e^{x}+e^{-x})^{2}}{4}-\frac{(e^{x}-e^{-x})^{2}}{4} \\ =&\ \frac{(e^{2x}+e^{-2x}+2)-(e^{2x}+e^{-x}-2)}{4} \\ =&\ \frac{4}{4} \\ =&\ 1 \end{align*}

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