Proving $\sinh{x}$ is strictly increasing over all the reals and $\cosh{x}$ is strictly decreasing on $(- \infty , 0]$

Question

How would I be proving $\sinh{x}$ is strictly increasing and $\cosh{x}$ is strictly decreasing on $(- \infty , 0]$

I succeded in showing $\cosh{x}$ is strictly increasing on the interval $[0, \infty)$ but having trouble with the rest. The $\sinh$ one is really messing with me.

*I can not use derivatives. I know how to but am not allowed - for some reason.

Answer 1

hint

For any real $ x$,

$$S(x)=\sinh(x)=\frac{e^x-e^{-x}}{2}$$

is the sum of two strictly increasing functions.