hyperbolic identities

There are many formulas involving hyperbolic functions, many of which are similar to formulas for trigonometric functions. Below is a list of some of these formulas (usually for real arguments).

1. Hyperbolic version of Pythagorean identities
   • $\cosh^2 x-\sinh^2 x=1$
   • $1-\tanh^2 x=\sech^2 x$
   • $\coth^2 x-1=\csch^2 x$
2. Fractional identities
   • $\ds \tanh x=\frac{\sinh x}{\cosh x}$
   • $\ds \coth x=\frac{\cosh x}{\sinh x}$
   • $\ds \coth x=\frac{1}{\tanh x}$
   • $\ds \tanh x=\frac{1}{\coth x}$
   • $\ds \csch x=\frac{1}{\sinh x}$
   • $\ds \sech x=\frac{1}{\cosh x}$
3. Hyperbolic functions of a purely imaginary number
   • $\sinh(ix)=i\sin x$
   • $\cosh(ix)=\cos x$
   • $\tanh(ix)=i\tan x$
   • $\cosh(ix)=i\cot x$
   • $\csch(ix)=i\csc x$
   • $\sech(ix)=\sec x$
4. Addition formulas and subtraction formulas
   • $\sinh(x\pm y)=\sinh x\cosh y\pm\cosh x\sinh y$
   • $\cosh(x\pm y)=\cosh x\cosh y\pm\sinh x\sinh y$
   • $\ds \tanh(x\pm y)=\frac{\tanh x\pm\tanh y}{1\pm\tanh x\tanh y}$
5. Formulas for hyperbolic functions of a complex number
   • $\sinh(x+iy)=\sinh x\cos y+i\cosh x\sin y$
   • $\cosh(x+iy)=\cosh x\cos y+i\sinh x\sin y$
   • $\ds \tanh(x+iy)=\frac{\tanh x+i\tan y}{1+i\tanh x\tan y}$
6. Opposite formulas
   • $\sinh(-x)=-\sinh x$
   • $\cosh(-x)=\cosh x$
   • $\tanh(-x)=-\tanh x$
7. Double argument formulas
   • $\sinh(2x)=2\sinh x\cosh x$
   • $\cosh(2x)=\cosh^2 x+\sinh^2 x=2\cosh^2 x-1=1+2\sinh^2 x$
   • $\ds \tanh(2x)=\frac{2\tanh x}{1+\tanh^2 x}$
8. Periodicity formulas
   • $\sinh(z+2\pi i) = \sinh{z}$
   • $\cosh(z+2\pi i) = \cosh{z}$
   • $\tanh(z+\pi i) = \tanh{z}$

   Cf. the periodicity of exponential function.

9. Exponential formulas
   • $\ds \cosh x=\frac{e^x+e^{-x}}{2}$
   • $\ds \sinh x=\frac{e^x-e^{-x}}{2}$
   • $\ds \tanh x=\frac{e^x-e^{-x}}{e^x+e^{-x}}$
   • $e^x=\cosh x+\sinh x$
   • $e^{-x}=\cosh x-\sinh x$

   Note that the first three formulas given in this section are definitions.