Solution
= Solution
For $x>0$, the exponential definitions of the <hyperbolic functions> give $\cosh x-\sinh x=e^{-x}>0$ and $\sinh x>0$. Taking reciprocals of these positive quantities proves $1/\cosh x<1/\sinh x$. Also $\cosh x>1$ for $x>0$, so division by the positive $\sinh x$ gives $1/\sinh x<\cosh x/\sinh x$. Combining,
$$
\boxed{\frac1{\cosh x}<\frac1{\sinh x}<\coth x\qquad(x>0)}.
$$