The value of $$\sqrt{\frac{\cosec \phi - \cot \phi}{\cosec \phi + \cot \phi}} \div \frac{\sin \phi}{1 + \cos \phi}$$ is equal to:

$$\sqrt{\frac{\cosec \phi - \cot \phi}{\cosec \phi + \cot \phi}} \div \frac{\sin \phi}{1 + \cos \phi}$$
= $$\sqrt{\frac{\cosec \phi - \cot \phi}{\cosec \phi + \cot \phi}} \times \frac{1 + \cos \phi}{\sin \phi}$$
= $$\sqrt{\frac{\cosec \phi - \cot \phi}{\cosec \phi + \cot \phi} \times \frac{\cosec \phi - \cot \phi}{\cosec \phi - \cot \phi}} \times \frac{1 + \cos \phi}{\sin \phi}$$
= $$\sqrt{\frac{(\cosec \phi - \cot \phi)^2}{\cosec^2 \phi - \cot^2 \phi}} \times \frac{1 + \cos \phi}{\sin \phi}$$
= $$(\cosec \phi - \cot \phi) \times \frac{1 + \cos \phi}{\sin \phi}$$
= $$(\cosec \phi - \cot \phi) \times (\cosec \phi + \cot \phi)$$
= $$\cosec^2 \phi - \cot^2 \phi$$ = 1