#### Answer

$cos~2A = \frac{cot~A-tan~A}{csc~A~sec~A}$

#### Work Step by Step

We can verify the identity:
$cos~2A = cos^2~A-sin^2~A$
$cos~2A = cos~A~cos~A-sin~A~sin~A$
$cos~2A = (cos~A~cos~A-sin~A~sin~A)(\frac{\frac{1}{sin~A~cos~A}}{\frac{1}{sin~A~cos~A}})$
$cos~2A = \frac{(\frac{cos~A}{sin~A}-\frac{sin~A}{cos~A})}{{\frac{1}{sin~A~cos~A}}}$
$cos~2A = \frac{cot~A-tan~A}{csc~A~sec~A}$