Answer
The identity is verified.
$\frac{cot^3t}{csc~t}$: black
$cos~t(csc^2t-1)$: red
Work Step by Step
$\frac{cot^3t}{csc~t}=\frac{\frac{cos^3t}{sin^3t}}{\frac{1}{sin~t}}=\frac{cos^3t}{sin^2t}=cos~t~\frac{cos^2t}{sin^2t}=cos~t~cot^2t=cos~t(csc^2t-1)$
