# Hyperbolic Functions Identities and Formulas

 
 Definitions of Hyperbolic Functions $\sinh x = \dfrac{e^x - e^{-x}}{2}$ $\cosh x = \dfrac{e^x + e^{-x}}{2}$ $\tanh x = \dfrac{e^x - e^{-x}}{e^x + e^{-x}}$ $\coth x = \dfrac{e^x + e^{-x}}{e^x - e^{-x}}$ $\text{sech} x = \dfrac{2}{e^x + e^{-x}}$ $\text{csch} x = \dfrac{2}{e^x - e^{-x}}$ Hyperbolic Functions Identities for Negative Arguments Odd Functions $\sinh (-x) = -\sinh x$ $\tanh (-x) = -\tanh x$ $\coth (-x) = -\coth x$ $\text{csch} (-x) = -\text{csch} x$ Even Functions $\cosh (-x) = \cosh x$ $\text{sech} (-x) = \text{sech} x$ Hyperbolic Functions Identities $\tanh x = \dfrac {\sinh x}{\cosh x}$ $\coth x = \dfrac {\cosh x}{\sinh x}$ $\text{sech} x = \dfrac {1}{\cosh x}$ $\text{csch} x = \dfrac {1}{\sinh x}$ $\cosh^2 x= 1+\sinh^2 x$ $\text{sech}^2 x + \tanh^2 x= 1$ $\coth^2 x = 1+ \text{csch}^2 x$ Hyperbolic Functions Addition of Argumemts Formulas $\sinh(x+y)=\sinh x \cosh y + \cosh x \sinh y$ $\sinh(x-y)=\sinh x \cosh y - \cosh x \sinh y$ $\cosh(x+y)=\cosh x \cosh y + \sinh x \sinh y$ $\cosh(x-y)=\cosh x \cosh y - \sinh x \sinh y$ $\tanh(x+y)=\dfrac{\tanh x + \tanh y}{1+\tanh x \cdot \tanh y}$ $\tanh(x-y)=\dfrac{\tanh x - \tanh y}{1-\tanh x \cdot \tanh y}$ $\coth(x+y)=\dfrac{\coth x \cdot \tanh y + 1}{\coth x + \coth y}$ $\coth(x-y)=\dfrac{\coth x \cdot \tanh y - 1}{\coth x - \coth y}$ Hyperbolic Functions Double Argumemt Formulas $\sinh 2x= 2\sinh x \cosh x$ $\cosh 2x= \cosh^2 x + \sinh^2 x = 2 \cosh^2 - 1 = 1+2\sinh^2 x$ $\tanh 2x =\dfrac{2 \tanh x}{1+\tanh^2 x}$ $\coth 2x =\dfrac{\coth^2 x+1}{2\coth x}$