1. **State the problem:** Prove that $$\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \csc A - \cot A$$. 2. **Recall the formulas:** - $$\csc A = \frac{1}{\sin A}$$ - $$\cot A = \frac{\cos A}{\sin A}$$ - Also, use the Pythagorean identity: $$\sin^2 A = 1 - \cos^2 A$$. 3. **Simplify the left side:** Start with $$\sqrt{\frac{1 - \cos A}{1 + \cos A}}$$. Multiply numerator and denominator inside the root by $$1 - \cos A$$ to rationalize the denominator: $$\sqrt{\frac{(1 - \cos A)^2}{(1 + \cos A)(1 - \cos A)}} = \sqrt{\frac{(1 - \cos A)^2}{1 - \cos^2 A}}$$. 4. **Use the Pythagorean identity:** Since $$1 - \cos^2 A = \sin^2 A$$, the expression becomes: $$\sqrt{\frac{(1 - \cos A)^2}{\sin^2 A}} = \frac{|1 - \cos A|}{|\sin A|}$$. 5. **Consider the domain:** For angles where $$\sin A > 0$$ and $$1 - \cos A \geq 0$$, we can drop the absolute values: $$\frac{1 - \cos A}{\sin A}$$. 6. **Rewrite the right side:** $$\csc A - \cot A = \frac{1}{\sin A} - \frac{\cos A}{\sin A} = \frac{1 - \cos A}{\sin A}$$. 7. **Conclusion:** Both sides equal $$\frac{1 - \cos A}{\sin A}$$, so the identity is proven: $$\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \csc A - \cot A$$.