Real analysis - Showing that Sinh(x) is strictly increasing with the use of logarithmic hyperbolic formulas

Question

I need to show

• that $\sinh{x}$ is strictly increasing on all of $\mathbb{R}$
• that $\cosh(x)$ is stricly increasing on $[0,\infty)$ and strictly decreasing on $(\infty,0]$

I need to do it through the use of the logarithmic hyperbolic formulas: \begin{align*} \sinh x-\sinh y=2 \cosh \frac{x+y}{2} \sinh \frac{x-y}{2}\\ \cosh x-\cosh y=2 \sinh \frac{x+y}{2} \sinh \frac{x-y}{2} \end{align*}

I also know

• $\cosh -x = \cosh x$
• $\sinh -x = -\sinh x$

which means it should suffice to show the property on the interval $[0,\infty)$

I have already proven this through the use of the hyperbolic addition formulas which made sense to me - however it was marked as incorrect as i have to use the 2 formulas above.I am not quite sure how to go about using them, and would be really grateful for any help.

Answer 1

If $y>x\geqslant0$, then$$\sinh(y)-\sinh(x)=2\cosh\left(\frac{x+y}2\right)\sinh\left(\frac{y-x}2\right)>0,$$since $\frac{x+y}2,\frac{y-x}2>0$ and $(\forall t\in(0,\infty)):\sinh(t),\cosh(t)>0$. So, $\sinh(y)>\sinh(x)$.