1. Datos del problema:
Simplificar la fracción compleja $N$.

2. Fórmulas/Propiedades:

• $\sin^2 A - \sin^2 B = \sin(A+B)\sin(A-B)$


• $\cos^2 x - \cos^2 y = \sin^2 y - \sin^2 x$

3. Desarrollo paso a paso:
Analizamos el denominador:
$$D = \sin^2 x + [\sin^2(x+y) - \sin^2 y] = \sin^2 x + [\sin(x+y+y)\sin(x+y-y)]$$
$$D = \sin^2 x + \sin(x+2y)\sin x = \sin x [\sin x + \sin(x+2y)]$$
Aplicando suma a producto: $\sin A + \sin B = 2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2})$:
$$D = \sin x [2 \sin(x+y) \cos y]$$

Analizamos el numerador:
$$U = \cos^2 x - \cos^2 y + \sin^2(x+y) = (\sin^2 y - \sin^2 x) + \sin^2(x+y)$$
$$U = \sin(y+x)\sin(y-x) + \sin^2(x+y) = \sin(x+y) [\sin(y-x) + \sin(x+y)]$$
Aplicando suma a producto:
$$U = \sin(x+y) [2 \sin y \cos x]$$

Sustituyendo en $N$:
$$N = \left( \frac{2 \sin(x+y) \sin y \cos x}{2 \sin x \sin(x+y) \cos y} \right) \frac{\tan x}{\tan y} = \left( \frac{\sin y \cos x}{\sin x \cos y} \right) \frac{\frac{\sin x}{\cos x}}{\frac{\sin y}{\cos y}}$$
$$N = \left( \frac{\tan y}{\tan x} \right) \frac{\tan x}{\tan y} = 1$$

4. Resultado final:
$$N = 1$$