91Ó°ÊÓ

(a) Prove that \(\arctan x+\arctan y=\arctan \frac{x+y}{1-x y}, \quad x y \neq 1\) (b) Use the formula in part (a) to show that \(\arctan \frac{1}{2}+\arctan \frac{1}{3}=\frac{\pi}{4}\)

Show that \(\arctan (\sinh x)=\arcsin (\tanh x)\)

From the vertex \((0, c)\) of the catenary \(y=c \cosh (x / c)\) a line \(L\) is drawn perpendicular to the tangent to the catenary at a point \(P\). Prove that the length of \(L\) intercepted by the axes is equal to the ordinate \(y\) of the point \(P\).

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Is the converse of the second part of Theorem \(5.7\) true? That is, if a function is one-to-one (and therefore has an inverse function), then must the function be strictly monotonic? If so, prove it. If not, give a counterexample.

Prove that the natural logarithmic function is one-to-one.