1. Datos del problema:

• $A + B = \pi$ (o $180^\circ$)

2. Fórmulas/Propiedades:

• $\text{sen}(180^\circ + \theta) = -\text{sen } \theta$


• $\tan(360^\circ + \theta) = \tan \theta$


• $\cos(90^\circ + \theta) = -\text{sen } \theta$


• $\cot(90^\circ + \theta) = -\tan \theta$

3. Desarrollo paso a paso:

• Analizamos cada término usando $A+B = 180^\circ$:

1. $\text{sen}(2A+B) = \text{sen}(A + (A+B)) = \text{sen}(A + 180^\circ) = -\text{sen } A$
2. $\tan(3B+2A) = \tan(B + 2(B+A)) = \tan(B + 360^\circ) = \tan B$
3. $\cot(\frac{A+3B}{2}) = \cot(\frac{A+B+2B}{2}) = \cot(\frac{180^\circ+2B}{2}) = \cot(90^\circ+B) = -\tan B$
4. $\cos(\frac{B+3A}{2}) = \cos(\frac{B+A+2A}{2}) = \cos(\frac{180^\circ+2A}{2}) = \cos(90^\circ+A) = -\text{sen } A $

• Sustituimos en $ F $:

$$F = \frac{(-\text{sen } A)(\tan B)}{(-\tan B)(-\text{sen } A)} = \frac{-\text{sen } A \cdot \tan B}{\text{sen } A \cdot \tan B} = -1$$

4. Resultado final:
$$F = -1$$