1. Desarrollo paso a paso:
De la igualdad dada:
$$4 = (\cot x - \cos x)(\cot x + \cos x) \implies 4 = \cot^2 x - \cos^2 x$$
Sabemos que $\cot^2 x - \cos^2 x = \frac{\cos^2 x}{\text{sen}^2 x} - \cos^2 x = \cos^2 x \left( \frac{1 - \text{sen}^2 x}{\text{sen}^2 x} \right) = \cos^2 x \cot^2 x$.
Entonces:
$$\cos^2 x \cot^2 x = 4 \implies \cos x \cot x = 2 \text{ (considerando valor positivo)}$$
$$\frac{\cos^2 x}{\text{sen } x} = 2 \implies \frac{1 - \text{sen}^2 x}{\text{sen } x} = 2 \implies \frac{1}{\text{sen } x} - \text{sen } x = 2$$
Esto es: $\csc x - \text{sen } x = 2$.

Buscamos $A = \csc x + \text{sen } x$. Usamos la identidad de Legendre:
$$(\csc x + \text{sen } x)^2 - (\csc x - \text{sen } x)^2 = 4 (\csc x)(\text{sen } x)$$
$$A^2 - (2)^2 = 4(1) \implies A^2 = 4 + 4 \implies A = \sqrt{8}$$
$$A = 2\sqrt{2}$$

2. Resultado final:
$$A = 2\sqrt{2}$$