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(section:uge2L-extra)=
# Extra (optional) exercises -- Short Day

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These exercises are optional extra material for students who have finished the exercises of the day and have the time and interest for more.

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## Extra exercise 1: Hyperbolic tangent

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In Extra Exercise 2 from this week's Long Day, the hyperbolic functions were introduced with the following expressions and both with domain and codomain $\mathbb R$: 

$$\mathrm{sinh}(x)=\frac{\mathrm e^x-\mathrm e^{-x}}{2}\quad\text{and}\mathrm{cosh}(x)=\frac{\mathrm e^x+\mathrm e^{-x}}{2}.$$

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### Question a

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The symbols $\mathrm{cosh}$ and $\mathrm{sinh}$ were chosen because of the resemblance between these hyperbolic functions and the better-known trigonometric functions $\cos$ and $\sin$. A famous identity, sometimes known as the "dummy rule", from the usual trigonometric functions is $\cos^2(x)+\sin^2(x)=1$. Show that the corresponding, albeit different, identity for the hyperbolic functions is $\mathrm{cosh}^2(x)-\mathrm{sinh}^2(x)=1$. 

Note that while the point $(\cos(x),\sin(x))$ lies on the unit circle, the point $(\mathrm{cosh}(x),\mathrm{sinh}(x))$ lies on a *hyperbola*, hence the name. 

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### Question b

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Inspired by the formula $\tan(x)=\sin(x)/\cos(x)$, we define $\mathrm{tanh}(x)=\mathrm{sinh}(x)/\mathrm{cosh}(x)$ for all $x \in \mathbb{R}$. The function $\mathrm{tanh}: \mathbb{R} \to \mathbb{R}$ defined by $x \mapsto \mathrm{tanh}(x)$ is called the hyperbolic tangent. What is the image set of this function?

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### Question c

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Show that $\mathrm{tanh}(x)=(\mathrm e^{2x}-1)/(\mathrm e^{2x}+1)$, and use this to find an expression for the inverse hyperbolic tangent function.


```{hint}
:class: dropdown
The function $\mathrm{tanh}: \mathbb{R} \to \mathbb{R}$ is not surjective and thus has no inverse function. However, if we restrict the codomain to the image set $B$ found in Question b, we obtain a function $\mathrm{tanh}: \mathbb{R} \to B$ that does have an inverse. It is this inverse you are looking for.
```