Chapter 5 - Review Exercises - Page 249: 58

$csc~A~sin~2A - sec~A = cos~2A~sec~A$

$csc~A~sin~2A - sec~A = \frac{2~sin~A~cos~A}{sin~A} - \frac{1}{cos~A}$ $csc~A~sin~2A - sec~A = 2~cos~A - \frac{1}{cos~A}$ $csc~A~sin~2A - sec~A = \frac{2~cos^2~A}{cos~A} - \frac{cos^2~A+sin^2~A}{cos~A}$ $csc~A~sin~2A - sec~A = \frac{cos^2~A -sin^2~A}{cos~A}$ $csc~A~sin~2A - sec~A = \frac{cos~2A}{cos~A}$ $csc~A~sin~2A - sec~A = cos~2A~sec~A$