Hyperbolic Functions Identities and Formulas

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Definitions of Hyperbolic Functions
  1. $\sinh x = \dfrac{e^x - e^{-x}}{2}$

  2. $\cosh x = \dfrac{e^x + e^{-x}}{2}$

  3. $\tanh x = \dfrac{e^x - e^{-x}}{e^x + e^{-x}}$

  4. $\coth x = \dfrac{e^x + e^{-x}}{e^x - e^{-x}}$

  5. $\text{sech} x = \dfrac{2}{e^x + e^{-x}}$

  6. $\text{csch} x = \dfrac{2}{e^x - e^{-x}}$
Hyperbolic Functions Identities for Negative Arguments
    Odd Functions

  1. $\sinh (-x) = -\sinh x$

  2. $\tanh (-x) = -\tanh x$

  3. $\coth (-x) = -\coth x$

  4. $\text{csch} (-x) = -\text{csch} x$

    Even Functions

  5. $\cosh (-x) = \cosh x$

  6. $\text{sech} (-x) = \text{sech} x$
Hyperbolic Functions Identities


  1. $\tanh x = \dfrac {\sinh x}{\cosh x}$

  2. $\coth x = \dfrac {\cosh x}{\sinh x}$

  3. $\text{sech} x = \dfrac {1}{\cosh x}$

  4. $\text{csch} x = \dfrac {1}{\sinh x}$

  5. $\cosh^2 x= 1+\sinh^2 x$

  6. $\text{sech}^2 x + \tanh^2 x= 1$

  7. $\coth^2 x = 1+ \text{csch}^2 x$
Hyperbolic Functions Addition of Argumemts Formulas


  1. $\sinh(x+y)=\sinh x \cosh y + \cosh x \sinh y$

  2. $\sinh(x-y)=\sinh x \cosh y - \cosh x \sinh y$

  3. $\cosh(x+y)=\cosh x \cosh y + \sinh x \sinh y$

  4. $\cosh(x-y)=\cosh x \cosh y - \sinh x \sinh y$

  5. $\tanh(x+y)=\dfrac{\tanh x + \tanh y}{1+\tanh x \cdot \tanh y}$

  6. $\tanh(x-y)=\dfrac{\tanh x - \tanh y}{1-\tanh x \cdot \tanh y}$

  7. $\coth(x+y)=\dfrac{\coth x \cdot \tanh y + 1}{\coth x + \coth y}$

  8. $\coth(x-y)=\dfrac{\coth x \cdot \tanh y - 1}{\coth x - \coth y}$
Hyperbolic Functions Double Argumemt Formulas


  1. $\sinh 2x= 2\sinh x \cosh x$

  2. $\cosh 2x= \cosh^2 x + \sinh^2 x = 2 \cosh^2 - 1 = 1+2\sinh^2 x$

  3. $\tanh 2x =\dfrac{2 \tanh x}{1+\tanh^2 x}$

  4. $\coth 2x =\dfrac{\coth^2 x+1}{2\coth x}$