# A calculus of the absurd

#### 16.2 Properties

##### 16.2.1 Odd/even nature

Note that as with their analogous trigonometric functions 117117 $$\sin (x)$$ is analogous to $$\sinh (x)$$ and $$\cos (x)$$ is analogous to $$\cosh (x)$$ , $$\sinh (x)$$ is an odd function, and $$\cosh (x)$$ is an even function.

[Proof that sinh(x) is an odd function]Proof that $$\sinh (x)$$ is an odd function

\begin{align} \label {proof that sinh is odd} \sinh (-x) & = \frac {e^{(-x)} - e^{-(-x)}}{2} \\ & = \frac {e^{(-x)} - e^{-(-x)}}{2} \\ & = \frac {e^{-x} - e^{x}}{2} \\ & = -\sinh (x) \end{align}

Proof that $$\cosh (x)$$ is an odd function

\begin{align} \label {Proof that cosh(x) is an odd function} \cosh (-x) & = \frac {e^{(-x)} + e^{-(-x)}}{2} \\ & = \frac {e^{-x} + e{x}}{2} \\ & = \cosh (x) \end{align}

Proof that $$\tanh (x)$$ is an odd function

118118 This can be proved by rewriting $$\tanh (x)$$ in terms of $$e^x$$, but as we’ve already proved the even/odd nature of $$\cos (x)$$ and $$\sin (x)$$ it makes sense to use that!

\begin{align} \label {Proof that tanh(x) is an odd function} \tanh (-x) & = \frac {\sinh (-x)}{\cosh (-x)} \\ & = \frac {-\sinh (x)}{\cosh (x)} \\ & = -\frac {\sinh (x)}{\cosh (x)} \\ & = -\tanh (x) \end{align}