1. Desarrollo:
Usamos la identidad $\sin^2 x - \sin^2 y = \sin(x+y)\sin(x-y)$ en los dos últimos términos:
$$ E = \sin^2 \frac{A}{2} + (\sin \frac{B}{2} + \sin \frac{C}{2})(\sin \frac{B}{2} - \sin \frac{C}{2}) $$
O alternativamente, usamos la forma simplificada para ángulos de un triángulo:
$$ \sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} = 1 - \cos^2 \frac{A}{2} + \sin^2 \frac{B}{2} $$
Usando identidades de degradación:
$$ E = \frac{1 - \cos A}{2} + \frac{1 - \cos B}{2} - \sin^2 \frac{C}{2} = 1 - \frac{1}{2}(\cos A + \cos B) - \sin^2 \frac{C}{2} $$
$$ E = 1 - \cos \frac{A+B}{2} \cos \frac{A-B}{2} - \sin^2 \frac{C}{2} $$
Dado $\frac{A+B}{2} = \frac{\pi}{2} - \frac{C}{2}$, entonces $\cos \frac{A+B}{2} = \sin \frac{C}{2}$:
$$ E = 1 - \sin \frac{C}{2} \cos \frac{A-B}{2} - \sin^2 \frac{C}{2} $$
Factorizando $-\sin \frac{C}{2}$:
$$ E = 1 - \sin \frac{C}{2} [ \cos \frac{A-B}{2} + \sin \frac{C}{2} ] $$
Sustituyendo $\sin \frac{C}{2} = \cos \frac{A+B}{2}$:
$$ E = 1 - \sin \frac{C}{2} [ \cos \frac{A-B}{2} + \cos \frac{A+B}{2} ] $$
Aplicando $\cos(x-y) + \cos(x+y) = 2 \cos x \cos y$:
$$ \boxed{\sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} - \sin^2 \frac{C}{2} = 1 - 2 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{C}{2}} $$