Considere el predicado p(x) : 2 〖sin〗 ^ 2⁡〖(x) = 1 - cos⁡(x) 〗 y x∈[0, π]La suma de los elementos de Ap(x) es :8π / 33 π4π / 34π7π / 3?

La suma de los elementos de Ap(x) nos da : 2π / 3Explicación : Tenemos la expresión : <img src="https://tex.z-dn.net/?f=2%5Csin%20%5Cleft%28x%5Cright%29%5E2%3D1-cos%5Cleft%28x%5Cright%29" />igualamos a cero<img src="https://tex.z-dn.net/?f=2%5Csin%20%5E2%5Cleft%28x%5Cright%29%2B%5Ccos%20%5Cleft%28x%5Cright%29-1%3D0" />Usamos la siguiente identidad : <img src="https://tex.z-dn.net/?f=%5Csin%20%5E2%5Cleft%28x%5Cright%29%3D1-%5Ccos%20%5E2%5Cleft%28x%5Cright%29" />Nos queda : <img src="https://tex.z-dn.net/?f=-1%2B%5Ccos%20%5Cleft%28x%5Cright%29%2B%5Cleft%281-%5Ccos%20%5E2%5Cleft%28x%5Cright%29%5Cright%29%5Ccdot%20%5C%3A2%3D0" />Simplificamos : <img src="https://tex.z-dn.net/?f=1%2B%5Ccos%20%5Cleft%28x%5Cright%29-2%5Ccos%20%5E2%5Cleft%28x%5Cright%29%3D0" />Realizamos un cambio de variable : <img src="https://tex.z-dn.net/?f=%5Ccos%20%5Cleft%28x%5Cright%29%3Du" /><img src="https://tex.z-dn.net/?f=1%2Bu-2u%5E2%3D0" />Al resolver nos queda : <img src="https://tex.z-dn.net/?f=u%3D-%5Cfrac%7B1%7D%7B2%7D%2C%5C%3Au%3D1" />Devolvemos el cambio de variables<img src="https://tex.z-dn.net/?f=%5Ccos%20%5Cleft%28x%5Cright%29%3D-%5Cfrac%7B1%7D%7B2%7D%2C%5C%3A%5Ccos%20%5Cleft%28x%5Cright%29%3D1" /><img src="https://tex.z-dn.net/?f=%5Ccos%20%5Cleft%28x%5Cright%29%3D-%5Cfrac%7B1%7D%7B2%7D%5Cquad%20%3A%5Cquad%20x%3D%5Cfrac%7B2%5Cpi%20%7D%7B3%7D%2B2%5Cpi%20n%2C%5C%3Ax%3D%5Cfrac%7B4%5Cpi%20%7D%7B3%7D%2B2%5Cpi%20n" />[img = 10]Como x∈[0, π] la única solución deseada es : x = 2π / 3 y x = 0Sumamos las dos soluciones : Suma de los elementos = 2π / 3 + 0 = 2π / 3.