Find $\sinh x$ in terms of $\tanh x$.

Question

Given that $\tanh(x) = u$, find an expression for $\sinh(x)$ in terms of $u$.

I don't really know what the question wants from me here. Any help would be great.

Answer 1

$\displaystyle\tanh(x)=u,$

$$\implies u^2=\frac{\sinh^2(x)}{\cosh^2(x)}=\frac{\sinh^2(x)}{1+\sinh^2(x)}$$

Rearrange and express $\sinh^2(x)$ in terms of $u$

As $\displaystyle\cosh(x)=\frac{e^x+e^{-x}}2\ge\sqrt{e^x\cdot e^{-x}}=1$ for real $x$

$\displaystyle\sinh(x),\tanh(x)$ must have same sign

Answer 2

$$u=\tanh(x)=\dfrac{\sinh(x)}{\cosh(x)}\Rightarrow \sinh(x)=\tanh(x)\cosh(x)=u\cosh(x).$$