$$\frac{\cot \theta}{(1 - \sin \theta)(\sec \theta + \tan \theta)}$$ is equal to:

$$= \frac{\cot \theta}{(1 - \sin \theta)(\sec \theta + \tan \theta)}$$

$$=\frac{\frac{\cos\ \theta}{\sin\ \theta}}{(1-\sin\theta)(\frac{1}{\cos\ \theta}+\frac{\sin\ \theta}{\cos\ \theta})}$$

$$=\frac{\frac{\cos\ \theta}{\sin\ \theta}}{(1-\sin\theta)\ \ \frac{(1+\sin\ \theta)}{\cos\ \theta}}$$

$$=\frac{\frac{\left(\cos\ \theta\right)^2}{\sin\ \theta}}{(1-\sin^2\theta)\ \ }$$

$$=\frac{\frac{\cos^2\theta}{\sin\ \theta}}{\cos^2\theta\ \ }$$

$$=\frac{\cos^2\theta}{\sin\ \theta\ \times\ \cos^2\theta}\ \ $$

$$=\frac{1}{\sin\ \theta\ }\ \ $$

$$=\operatorname{cosec}\ \theta\ \ \ $$