# §4.35 Identities

###### Contents

 4.35.1 $\displaystyle\sinh\left(u\pm v\right)$ $\displaystyle=\sinh u\cosh v\pm\cosh u\sinh v,$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.24 (modified) Permalink: http://dlmf.nist.gov/4.35.E1 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4 4.35.2 $\displaystyle\cosh\left(u\pm v\right)$ $\displaystyle=\cosh u\cosh v\pm\sinh u\sinh v,$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.25 (modified) Permalink: http://dlmf.nist.gov/4.35.E2 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4 4.35.3 $\displaystyle\tanh\left(u\pm v\right)$ $\displaystyle=\frac{\tanh u\pm\tanh v}{1\pm\tanh u\tanh v},$ ⓘ Symbols: $\tanh\NVar{z}$: hyperbolic tangent function A&S Ref: 4.5.26 (modified) Permalink: http://dlmf.nist.gov/4.35.E3 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4 4.35.4 $\displaystyle\coth\left(u\pm v\right)$ $\displaystyle=\frac{\pm\coth u\coth v+1}{\coth u\pm\coth v}.$ ⓘ Symbols: $\coth\NVar{z}$: hyperbolic cotangent function A&S Ref: 4.5.27 (modified) Permalink: http://dlmf.nist.gov/4.35.E4 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4
 4.35.5 $\displaystyle\sinh u+\sinh v$ $\displaystyle=2\sinh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right),$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.41 Permalink: http://dlmf.nist.gov/4.35.E5 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4 4.35.6 $\displaystyle\sinh u-\sinh v$ $\displaystyle=2\cosh\left(\frac{u+v}{2}\right)\sinh\left(\frac{u-v}{2}\right),$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.42 Permalink: http://dlmf.nist.gov/4.35.E6 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4 4.35.7 $\displaystyle\cosh u+\cosh v$ $\displaystyle=2\cosh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right),$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function A&S Ref: 4.5.43 Permalink: http://dlmf.nist.gov/4.35.E7 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4 4.35.8 $\displaystyle\cosh u-\cosh v$ $\displaystyle=2\sinh\left(\frac{u+v}{2}\right)\sinh\left(\frac{u-v}{2}\right),$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.44 Permalink: http://dlmf.nist.gov/4.35.E8 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4 4.35.9 $\displaystyle\tanh u\pm\tanh v$ $\displaystyle=\frac{\sinh\left(u\pm v\right)}{\cosh u\cosh v},$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function and $\tanh\NVar{z}$: hyperbolic tangent function A&S Ref: 4.5.45 Permalink: http://dlmf.nist.gov/4.35.E9 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4 4.35.10 $\displaystyle\coth u\pm\coth v$ $\displaystyle=\frac{\sinh\left(v\pm u\right)}{\sinh u\sinh v}.$ ⓘ Symbols: $\coth\NVar{z}$: hyperbolic cotangent function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.46 (modified) Permalink: http://dlmf.nist.gov/4.35.E10 Encodings: TeX, pMML, png See also: Annotations for 4.35(i), 4.35 and 4

## §4.35(ii) Squares and Products

 4.35.11 ${\cosh^{2}}z-{\sinh^{2}}z=1,$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.16 Permalink: http://dlmf.nist.gov/4.35.E11 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii), 4.35 and 4
 4.35.12 ${\operatorname{sech}^{2}}z=1-{\tanh^{2}}z,$ ⓘ Symbols: $\operatorname{sech}\NVar{z}$: hyperbolic secant function, $\tanh\NVar{z}$: hyperbolic tangent function and $z$: complex variable A&S Ref: 4.5.17 Permalink: http://dlmf.nist.gov/4.35.E12 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii), 4.35 and 4
 4.35.13 ${\operatorname{csch}^{2}}z={\coth^{2}}z-1.$ ⓘ Symbols: $\operatorname{csch}\NVar{z}$: hyperbolic cosecant function, $\coth\NVar{z}$: hyperbolic cotangent function and $z$: complex variable A&S Ref: 4.5.18 Permalink: http://dlmf.nist.gov/4.35.E13 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii), 4.35 and 4
 4.35.14 $\displaystyle 2\sinh u\sinh v$ $\displaystyle=\cosh\left(u+v\right)-\cosh\left(u-v\right),$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.38 Permalink: http://dlmf.nist.gov/4.35.E14 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii), 4.35 and 4 4.35.15 $\displaystyle 2\cosh u\cosh v$ $\displaystyle=\cosh\left(u+v\right)+\cosh\left(u-v\right),$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function A&S Ref: 4.5.39 Permalink: http://dlmf.nist.gov/4.35.E15 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii), 4.35 and 4 4.35.16 $\displaystyle 2\sinh u\cosh v$ $\displaystyle=\sinh\left(u+v\right)+\sinh\left(u-v\right).$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.40 Permalink: http://dlmf.nist.gov/4.35.E16 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii), 4.35 and 4
 4.35.17 $\displaystyle{\sinh^{2}}u-{\sinh^{2}}v$ $\displaystyle=\sinh\left(u+v\right)\sinh\left(u-v\right),$ ⓘ Symbols: $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.47 Permalink: http://dlmf.nist.gov/4.35.E17 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii), 4.35 and 4 4.35.18 $\displaystyle{\cosh^{2}}u-{\cosh^{2}}v$ $\displaystyle=\sinh\left(u+v\right)\sinh\left(u-v\right),$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.47 Permalink: http://dlmf.nist.gov/4.35.E18 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii), 4.35 and 4 4.35.19 $\displaystyle{\sinh^{2}}u+{\cosh^{2}}v$ $\displaystyle=\cosh\left(u+v\right)\cosh\left(u-v\right).$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $\sinh\NVar{z}$: hyperbolic sine function A&S Ref: 4.5.48 Permalink: http://dlmf.nist.gov/4.35.E19 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii), 4.35 and 4

## §4.35(iii) Multiples of the Argument

 4.35.20 $\sinh\frac{z}{2}=\left(\frac{\cosh z-1}{2}\right)^{1/2},$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.28 Permalink: http://dlmf.nist.gov/4.35.E20 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4
 4.35.21 $\cosh\frac{z}{2}=\left(\frac{\cosh z+1}{2}\right)^{1/2},$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $z$: complex variable A&S Ref: 4.5.29 Permalink: http://dlmf.nist.gov/4.35.E21 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4
 4.35.22 $\tanh\frac{z}{2}=\left(\frac{\cosh z-1}{\cosh z+1}\right)^{1/2}=\frac{\cosh z-% 1}{\sinh z}=\frac{\sinh z}{\cosh z+1}.$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function, $\tanh\NVar{z}$: hyperbolic tangent function and $z$: complex variable A&S Ref: 4.5.30 Permalink: http://dlmf.nist.gov/4.35.E22 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4

The square roots assume their principal value on the positive real axis, and are determined by continuity elsewhere.

 4.35.23 $\displaystyle\sinh\left(-z\right)$ $\displaystyle=-\sinh z,$ ⓘ Symbols: $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.21 Permalink: http://dlmf.nist.gov/4.35.E23 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4 4.35.24 $\displaystyle\cosh\left(-z\right)$ $\displaystyle=\cosh z,$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $z$: complex variable A&S Ref: 4.5.22 Permalink: http://dlmf.nist.gov/4.35.E24 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4 4.35.25 $\displaystyle\tanh\left(-z\right)$ $\displaystyle=-\tanh z.$ ⓘ Symbols: $\tanh\NVar{z}$: hyperbolic tangent function and $z$: complex variable A&S Ref: 4.5.23 Permalink: http://dlmf.nist.gov/4.35.E25 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4
 4.35.26 $\sinh\left(2z\right)=2\sinh z\cosh z=\frac{2\tanh z}{1-{\tanh^{2}}z},$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function, $\tanh\NVar{z}$: hyperbolic tangent function and $z$: complex variable A&S Ref: 4.5.31 Permalink: http://dlmf.nist.gov/4.35.E26 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4
 4.35.27 $\cosh\left(2z\right)=2{\cosh^{2}}z-1=2{\sinh^{2}}z+1\\ ={\cosh^{2}}z+{\sinh^{2}}z,$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.32 Permalink: http://dlmf.nist.gov/4.35.E27 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4
 4.35.28 $\tanh\left(2z\right)=\frac{2\tanh z}{1+{\tanh^{2}}z},$ ⓘ Symbols: $\tanh\NVar{z}$: hyperbolic tangent function and $z$: complex variable A&S Ref: 4.5.33 Permalink: http://dlmf.nist.gov/4.35.E28 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4
 4.35.29 $\sinh\left(3z\right)=3\sinh z+4{\sinh^{3}}z,$ ⓘ Symbols: $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.34 Permalink: http://dlmf.nist.gov/4.35.E29 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4
 4.35.30 $\cosh\left(3z\right)=-3\cosh z+4{\cosh^{3}}z,$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $z$: complex variable A&S Ref: 4.5.35 Permalink: http://dlmf.nist.gov/4.35.E30 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4
 4.35.31 $\displaystyle\sinh\left(4z\right)$ $\displaystyle=4{\sinh^{3}}z\cosh z+4{\cosh^{3}}z\sinh z,$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.36 Permalink: http://dlmf.nist.gov/4.35.E31 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4 4.35.32 $\displaystyle\cosh\left(4z\right)$ $\displaystyle={\cosh^{4}}z+6{\sinh^{2}}z{\cosh^{2}}z+{\sinh^{4}}z.$ ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.37 Permalink: http://dlmf.nist.gov/4.35.E32 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii), 4.35 and 4
 4.35.33 $\cosh\left(nz\right)\pm\sinh\left(nz\right)=(\cosh z\pm\sinh z)^{n},$ $n\in\mathbb{Z}$.

## §4.35(iv) Real and Imaginary Parts; Moduli

With $z=x+iy$

 4.35.34 $\displaystyle\sinh z$ $\displaystyle=\sinh x\cos y+i\cosh x\sin y,$ 4.35.35 $\displaystyle\cosh z$ $\displaystyle=\cosh x\cos y+i\sinh x\sin y,$ 4.35.36 $\displaystyle\tanh z$ $\displaystyle=\frac{\sinh\left(2x\right)+i\sin\left(2y\right)}{\cosh\left(2x% \right)+\cos\left(2y\right)},$ 4.35.37 $\displaystyle\coth z$ $\displaystyle=\frac{\sinh\left(2x\right)-i\sin\left(2y\right)}{\cosh\left(2x% \right)-\cos\left(2y\right)}.$
 4.35.38 $|\sinh z|=({\sinh^{2}}x+{\sin^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2x% \right)-\cos\left(2y\right))\right)^{1/2},$
 4.35.39 $|\cosh z|=({\sinh^{2}}x+{\cos^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2x% \right)+\cos\left(2y\right))\right)^{1/2},$
 4.35.40 $|\tanh z|=\left(\frac{\cosh\left(2x\right)-\cos\left(2y\right)}{\cosh\left(2x% \right)+\cos\left(2y\right)}\right)^{1/2}.$