1. Identidades:
$2\cos \alpha \cos \beta = \cos(\alpha + \beta) + \cos(\alpha - \beta)$.

2. Desarrollo:
Sustituimos en la expresión:
$$E = \cos^2(\alpha + \beta) - [\cos(\alpha + \beta) + \cos(\alpha - \beta)]\cos(\alpha + \beta) + \cos^2 \beta$$
$$E = \cos^2(\alpha + \beta) - \cos^2(\alpha + \beta) - \cos(\alpha - \beta)\cos(\alpha + \beta) + \cos^2 \beta$$
$$E = \cos^2 \beta - \cos(\alpha + \beta)\cos(\alpha - \beta)$$

Usamos la propiedad: $\cos(A+B)\cos(A-B) = \cos^2 A - \sin^2 B$.
$$E = \cos^2 \beta - (\cos^2 \alpha - \sin^2 \beta)$$
$$E = \cos^2 \beta + \sin^2 \beta - \cos^2 \alpha$$
$$E = 1 - \cos^2 \alpha$$

3. Resultado final:
$$E = \sin^2 \alpha$$