Formulas and Results of Hyperbolic Functions

    • \[Sinhx = \frac{{{e^x} – {e^{ – x}}}}{2}\]
    • \[Coshx = \frac{{{e^x} + {e^{ – x}}}}{2}\]
    • \[Tanhx = \frac{{{e^x} – {e^{ – x}}}}{{{e^x} + {e^{ – x}}}}\]
    • \[Co\sec hx = \frac{2}{{{e^x} – {e^{ – x}}}}\]
    • \[Sechx = \frac{2}{{{e^x} + {e^{ – x}}}}\]
    • \[Cothx = \frac{{{e^x} + {e^{ – x}}}}{{{e^x} – {e^{ – x}}}}\]
    • \[Sinh( – x) = – Sinhx\]
    • \[Cosh( – x) = Coshx\]
    • \[Tanh( – x) = – Tanhx\]
    • \[Coth( – x) = – Cothx\]
    • \[Sech( – x) = Sechx\]
    • \[Co\sec h( – x) = – Co\sec hx\]
    • \[Cos{h^2}x – Sin{h^2}x = 1\]
    • \[Tan{h^2}x + Sec{h^2}x = 1\]
    • \[Cot{h^2}x – Co\sec {h^2}x = 1\]
    • \[Sin{h^2}x = \frac{{Cosh2x – 1}}{2}\]
    • \[Cos{h^2}x = \frac{{Cosh2x + 1}}{2}\]
    • \[Co\sec {h^2}x – Sec{h^2}x = Co\sec {h^2}x{\text{ }}Sec{h^2}x\]
    • \[Sinh(x + y) = Sinhx{\text{ }}Coshy + Coshx{\text{ }}Sinhy\]
    • \[Sinh(x – y) = Sinhx{\text{ }}Coshy + Coshx{\text{ }}Sinhy\]
    • \[Cosh(x + y) = Coshx{\text{ }}Coshy + Sinhx{\text{ }}Sinhy\]
    • \[Cosh(x – y) = Coshx{\text{ }}Coshy – Sinhx{\text{ }}Sinhy\]
    • \[Tanh(x + y) = \frac{{Tanhx – Tanhy}}{{1 + Tanhx{\text{ }}Tanhy}}\]
    • \[Tanh(x – y) = \frac{{Tanhx + Tanhy}}{{1 – Tanhx{\text{ }}Tanhy}}\]
    • \[Sinh(x + y){\text{ }}Sinh(x – y) = Sin{h^2}x – Sin{h^2}y = Cos{h^2}x – Cos{h^2}y\]
    • \[Cosh(x + y){\text{ }}Cosh(x – y) = Sin{h^2}x + Cos{h^2}y = Cos{h^2}x + Sin{h^2}y\]
    • \[Sinhx + Sinhy = 2Sinh\left( {\frac{{x + y}}{2}} \right)Cosh\left( {\frac{{x – y}}{2}} \right)\]
    • \[Sinhx – Sinhy = 2Cosh\left( {\frac{{x + y}}{2}} \right)Sinh\left( {\frac{{x – y}}{2}} \right)\]
    • \[Coshx + Coshy = 2Cosh\left( {\frac{{x + y}}{2}} \right)Cosh\left( {\frac{{x – y}}{2}} \right)\]
    • \[Coshx – Coshy = 2Sinh\left( {\frac{{x + y}}{2}} \right)Sinh\left( {\frac{{x – y}}{2}} \right)\]
    • \[Sinh2x = 2{\text{ }}Sinhx{\text{ }}Coshx\]
    • \[Cosh2x = Cos{h^2}x + Sin{h^2}x = 2Cos{h^2}x – 1 = 1 + 2Sin{h^2}x\]
    • \[Tanh2x = \frac{{2Tanhx}}{{1 + Tan{h^2}x}}\]
    • \[Sinh3x = 3Sinhx + 4Sin{h^3}x\]
    • \[Cosh3x = 4Cos{h^3}x – 3Coshx\]
    • \[Tanh3x = \frac{{3Tanhx + Tan{h^3}x}}{{1 + Tan{h^2}x}}\]
    • \[Sinh\frac{x}{2} = \pm \sqrt {\frac{{Coshx – 1}}{2}} \]
    • \[Cosh\frac{x}{2} = \pm \sqrt {\frac{{Coshx + 1}}{2}} \]
    • \[Tanh\frac{x}{2} = \frac{{Coshx – 1}}{{Sinhx}} = \frac{{Sinhx}}{{1 + Coshx}}\]
    • \[Sinhx = 2Sinh\frac{x}{2}Cosh\frac{x}{2}\]
    • \[Coshx = 2Cos{h^2}\frac{x}{2} – 1 = Cos{h^2}\frac{x}{2} + Sin{h^2}\frac{x}{2} = 1 + 2Sin{h^2}\frac{x}{2}\]