Chapter 7.1, Problem 32ES

Derive the identity

$\frac{{\mathrm{sech}}^{2}x}{1+{\tanh }^{2}x}=\mathrm{sech}2x$

(b) Use the result in part (a) to evaluate ${\displaystyle \int \mathrm{sech}xdx.}$

(c) Derive the identity

$\mathrm{sech}x=\frac{2e^x}{e^{2x}+1}$

(d) Use the results in part (c) to evaluate ${\displaystyle \int \sec xdx.}$

(e) Explain why your to parts (b) and (d) are consistent.