Answer
The identity is verified.
$\frac{cos~\theta~cot~\theta}{1-sin~\theta}-1=csc~\theta$
Work Step by Step
$\frac{cos~\theta~cot~\theta}{1-sin~\theta}-1=\frac{cos~\theta~\frac{cos~\theta}{sin~\theta}}{1-sin~\theta}-\frac{1-sin~\theta}{1-sin~\theta}=\frac{\frac{cos^2\theta}{sin~\theta}-1+sin~\theta}{1-sin~\theta}\frac{sin~\theta}{sin~\theta}=\frac{cos^2\theta-sin~\theta+sin^2\theta}{sin~\theta(1-sin~\theta)}=\frac{1-sin~\theta}{sin~\theta(1-sin~\theta)}=\frac{1}{sin~\theta}=csc~\theta$
