Putnam 1979 B1

Problem B-1 Prove or disprove: there is at least one straight line normal to the graph of $y=\cosh x$ at a point $(a, \cosh a)$ and also normal to the graph of $y=\sinh x$ at a point $(c, \sinh c)$. [At a point on a graph, the normal line is the perpendicular to the tangent at that point. Also, $\cosh x=\left(e^{x}+\right.$ $\left.e^{-x}\right) / 2$ and $\left.\sinh x=\left(e^{x}-e^{-x}\right) / 2.\right]$