Hyperbolic Function Identities

Identities can be easily derived from the definitions.

$$\cosh^2 x - \sinh^2 x = 1\,$$

$$\sinh(-x) = -\sinh x\,\!$$

$$\cosh(-x) = \cosh x\,\!$$

$$\tanh(-x) = -\tanh x\,\!$$

$$\coth(-x) = -\coth x\,\!$$

The derivatives of the hyperbolic functions.

$$\frac{d}{dx}\sinh(x) = \cosh(x) \,$$

$$\frac{d}{dx}\cosh(x) = \sinh(x) \,$$

$$\frac{d}{dx}\tanh(x) = 1 - \tanh^2(x) = \hbox{sech}^2(x) = 1/\cosh^2(x) \,$$

$$\frac{d}{dx}\coth(x) = 1 - \coth^2(x) = -\hbox{csch}^2(x) = -1/\sinh^2(x) \,$$

Hyperbolic functions of sums.

$$\sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y \,$$

$$\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \,$$

$$\tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} \,$$

$$\sinh 2x\ = 2\sinh x \cosh x \,$$

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

$$\cosh^2\frac{x}{2} = \frac{\cosh x + 1}{2}$$

$$\sinh^2\frac{x}{2} = \frac{\cosh x - 1}{2}$$

Inverse hyperbolic functions from logs.

$$\coth ^{-1}x=\tanh ^{-1}\left( \frac{1}{x} \right)$$

$$\sinh ^{-1}x=\ln \left( x+\sqrt{x^{2}+1} \right)$$

$$\cosh ^{-1}x=\ln \left( x+\sqrt{x^{2}-1} \right);x\ge 1$$

$$\tanh ^{-1}x=\frac{1}{2}\ln \left( \frac{1+x}{1-x} \right);\left\vert x \right\vert<1$$

Hyperbolic sine and cosine are related to sine and cosine of imaginary numbers.

$$e^{i x} = \cos x + i \;\sin x$$

$$e^{-i x} = \cos x - i \;\sin x$$

$$\cosh ix = \frac{e^{i x} + e^{-i x}}{2} = \cos x$$

$$\sinh ix = \frac{e^{i x} - e^{-i x}}{2} = i \sin x$$

$$\tanh ix = i \tan x \,$$

$$\cosh x = \cos ix \,$$

$$\sinh x = -i \sin ix \,$$

$$\tanh x = -i \tan ix \,$$