Hyperbolic Identities

See also Trigonometric Identities.

Instruction of hyperbolic functions in math classes in precalc and calculus classes is rare. Nonetheless, this reference page will prove to be useful should you require them. Let $x$, $\alpha$ (alpha), and $\beta$ (beta) be arbitrary values (see Greek letters).

Basic Relationships

$$\begin{array}{rlr}\sinh x& =\frac{e^x-e^{-x}}{2}& \text{(hyperbolic sine)}\\ \cosh x& =\frac{e^x+e^{-x}}{2}& \text{(hyperbolic cosine)}\\ \tanh x& =\frac{\sinh x}{\cosh x}=\frac{1}{\coth x}& \text{(hyperbolic tangent)}\\ \coth x& =\frac{\cosh x}{\sinh x}=\frac{1}{\tanh x}& \text{(hyperbolic cotangent)}\\ \text{sech }x& =\frac{1}{\cosh x}& \text{(hyperbolic secant)}\\ \text{csch }x& =\frac{1}{\sinh x}& \text{(hyperbolic cosecant)}\end{array}$$

Pythagorean Identities

$$\begin{array}{c}{\cosh }^{2}x-{\sinh }^{2}x=1\\ {\tanh }^{2}x+{\text{sech}}^{2}x=1\\ {\text{coth}}^{2}x-{\text{csch}}^{2}x=1\end{array}$$

Odd/Even Identities

$$\begin{array}{rlrl}& \sinh (-x)& & =-\sinh x\\ & \cosh (-x)& & =\cosh x\\ & \tanh (-x)& & =-\tanh x\\ & \text{sech}(-x)& & =\text{sech }x\\ & \text{csch}(-x)& & =-\text{csch }x\\ & \text{coth}(-x)& & =-\text{coth }x\end{array}$$

Sum and Difference Formulas

$$\begin{array}{rl}\sinh (\alpha \pm \beta )& =\sinh \alpha \cosh \beta \pm \cosh \alpha \sinh \beta \\ \cosh (\alpha \pm \beta )& =\cosh \alpha \cosh \beta \pm \sinh \alpha \sinh \beta \\ \tanh (\alpha \pm \beta )& =\frac{\tanh \alpha \pm \tanh \beta }{1\pm \tanh \alpha \tanh \beta }\end{array}$$

Double-Value Formulas

$$\begin{array}{rl}\sinh 2x& =2\sinh x\cosh x\\ \cosh 2x& ={\cosh }^{2}x+{\sinh }^{2}x\\ & =1+2{\sinh }^{2}x\\ & =2{\cosh }^{2}x-1\\ \tanh 2x& =\frac{2\tanh x}{1+{\tanh }^{2}x}\end{array}$$

Half-Value Formulas

$$\begin{array}{c}\sinh \frac{x}{2}=\pm \sqrt{\frac{\cosh x-1}{2}}\\ \cosh \frac{x}{2}=\sqrt{\frac{\cosh x+1}{2}}\\ \tanh \frac{x}{2}=\pm \sqrt{\frac{\cosh x-1}{\cosh x+1}}\end{array}$$

For $\sinh \frac{x}{2}$ and $\tanh \frac{x}{2}$, the sign is chosen to match the sign of $x$. No sign choice is needed for $\cosh \frac{x}{2}$, which is always positive.