Chapter 7.1, Problem 33ES

(a) Derive the identity

$\frac{{\sec }^{2}x}{\tan x}=\frac{1}{\sin x\cos x}$

(b) Use the identity $\sin 2x=2\sin x\cos x$ along with the results in part (a) to evaluate  ${\displaystyle \int \csc xdx.}$

(c) Use the identity $\cos x=\sin [(\pi /2)-x]$ along with your answer to part (a) to evaluate ${\displaystyle \int \sec xdx.}$