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

$$\sec\theta\ =\frac{1}{\cos\theta\ }\ ;\tan\theta\ =\frac{\sin\theta\ }{\cos\theta\ };\operatorname{cosec}\theta=\frac{1}{\sin\theta\ \ };\cot\theta=\frac{\cos\theta\ }{\sin\theta\ }\ \ \ $$
substituting we get :$$\frac{\left(1+\sin\theta\ \right)\left(1-\sin\theta\ \right)}{\frac{\cos\theta\ }{\frac{\left(1+\cos\theta\ \right)\left(1-\cos\theta\ \right)}{\sin\theta\ }\times\ \frac{1}{\sin\theta\ }}\ }$$
we get $$\frac{\frac{\left(1-\sin^2\theta\ \right)}{\cos\theta\ }}{\frac{\sin^2\theta}{\sin^2\theta\ }\ }$$
we get $$\cos\theta\ $$