<s>[INST] Prove $\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A$ by the Pythagorean theorem. How do I use the Pythagorean Theorem to prove  that $$\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A?$$ 
 [/INST] Let us denote base of right angle triangle as b, perpendicular ( height ) as p, and hypotenuse as h, 
$$\cos A = \frac{b}{h} ; \qquad \sin A = \frac{p}{h}\tag{i}$$
Therefore, $$\frac{\cos^2A}{1-\sin A} = \frac{\frac{b^2}{h^2}}{1-\frac{p}{h}}$$
[By putting the values of $\cos A$ and $\sin A$ from $(i)$]
Which after simplification gives you:
$$\frac{b^2}{h(h-p)}\tag{ii}$$
Now as
$$b^2 = h^2-p^2\quad\text{[using pythagoreous theorem]}\tag{iii}$$
By putting the value of $b^2$ from (iii) in (ii) you get : 
$$\frac{h^2-p^2}{h(h-p)} = \frac{h+p}{h} = 1+ \frac{p}{h} = 1+ \sin A$$
(hence proved)
</s>