# Formulas and Results of Hyperbolic Functions

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