The six hyperbolic functions and their natural domains

For $x\in\mathbb R$, define $\sinh x:=\frac{\exp x-\exp(-x)}2,\qquad\cosh x:=\frac{\exp x+\exp(-x)}2,$ $\tanh x:=\frac{\sinh x}{\cosh x},\qquad\operatorname{sech}x:=\frac1{\cosh x}.$ For $x\ne0$, define $\coth x:=\frac{\cosh x}{\sinh x},\qquad\operatorname{csch}x:=\frac1{\sinh x}.$

The positivity of $\exp$ makes $\cosh x>0$ for every $x$. If $x\ne0$, then $x\ne-x$, so strict increase of $\exp$ makes $\exp x\ne\exp(-x)$; hence $\sinh x\ne0$. Thus all four quotients above are defined on exactly their stated domains (The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$, The exponential function is strictly increasing).