Formulas and Results of Hyperbolic Functions

    • Sinhx =  \frac{{{e^x} - {e^{ - x}}}}{2}

    • Coshx =  \frac{{{e^x} + {e^{ - x}}}}{2}

    • Tanhx =  \frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}}

    • Co\sec hx =  \frac{2}{{{e^x} - {e^{ - x}}}}

    • Sechx =  \frac{2}{{{e^x} + {e^{ - x}}}}

    • Cothx =  \frac{{{e^x} + {e^{ - x}}}}{{{e^x} - {e^{ - x}}}}

    • Sinh( - x) = - Sinhx

    • Cosh( - x) =  Coshx

    • Tanh( - x) = - Tanhx

    • Coth( - x) = - Cothx

    • Sech( - x) = Sechx

    • Co\sec h( - x) = - Co\sec hx

    • Cos{h^2}x - Sin{h^2}x = 1

    • Tan{h^2}x + Sec{h^2}x = 1

    • Cot{h^2}x - Co\sec {h^2}x = 1

    • Sin{h^2}x = \frac{{Cosh2x - 1}}{2}

    • Cos{h^2}x = \frac{{Cosh2x + 1}}{2}

    • Co\sec {h^2}x - Sec{h^2}x = Co\sec {h^2}x{\text{  }}Sec{h^2}x

    • Sinh(x + y) = Sinhx{\text{ }}Coshy + Coshx{\text{  }}Sinhy

    • Sinh(x - y) = Sinhx{\text{ }}Coshy + Coshx{\text{  }}Sinhy

    • Cosh(x + y) = Coshx{\text{ }}Coshy + Sinhx{\text{  }}Sinhy

    • Cosh(x - y) = Coshx{\text{ }}Coshy - Sinhx{\text{  }}Sinhy

    • Tanh(x + y) = \frac{{Tanhx - Tanhy}}{{1 +  Tanhx{\text{ }}Tanhy}}

    • Tanh(x - y) = \frac{{Tanhx + Tanhy}}{{1 -  Tanhx{\text{ }}Tanhy}}

    • Sinh(x + y){\text{ }}Sinh(x - y) = Sin{h^2}x -  Sin{h^2}y = Cos{h^2}x - Cos{h^2}y

    • Cosh(x + y){\text{ }}Cosh(x - y) = Sin{h^2}x +  Cos{h^2}y = Cos{h^2}x + Sin{h^2}y

    • Sinhx + Sinhy = 2Sinh\left( {\frac{{x + y}}{2}}  \right)Cosh\left( {\frac{{x - y}}{2}} \right)

    • Sinhx - Sinhy = 2Cosh\left( {\frac{{x + y}}{2}}  \right)Sinh\left( {\frac{{x - y}}{2}} \right)

    • Coshx + Coshy = 2Cosh\left( {\frac{{x + y}}{2}}  \right)Cosh\left( {\frac{{x - y}}{2}} \right)

    • Coshx - Coshy = 2Sinh\left( {\frac{{x + y}}{2}}  \right)Sinh\left( {\frac{{x - y}}{2}} \right)

    • Sinh2x = 2{\text{ }}Sinhx{\text{ }}Coshx

    • Cosh2x = Cos{h^2}x + Sin{h^2}x = 2Cos{h^2}x - 1 = 1  + 2Sin{h^2}x

    • Tanh2x = \frac{{2Tanhx}}{{1 + Tan{h^2}x}}

    • Sinh3x = 3Sinhx + 4Sin{h^3}x

    • Cosh3x = 4Cos{h^3}x - 3Coshx

    • Tanh3x = \frac{{3Tanhx + Tan{h^3}x}}{{1 +  Tan{h^2}x}}

    • Sinh\frac{x}{2} =  \pm \sqrt {\frac{{Coshx - 1}}{2}}

    • Cosh\frac{x}{2} =  \pm \sqrt {\frac{{Coshx + 1}}{2}}

    • Tanh\frac{x}{2} = \frac{{Coshx - 1}}{{Sinhx}} =  \frac{{Sinhx}}{{1 + Coshx}}

    • Sinhx = 2Sinh\frac{x}{2}Cosh\frac{x}{2}

    • Coshx = 2Cos{h^2}\frac{x}{2} - 1 =  Cos{h^2}\frac{x}{2} + Sin{h^2}\frac{x}{2} = 1 + 2Sin{h^2}\frac{x}{2}