If x= y, then sin(x)+ sin(y)+ sin(x+ y)= sin(x)+ sin(x)+ sin(x+x)= 2sin(x)+ sin(2x).
I suggest you check that again. The values of x and y are NOT values of the function and cannot be maximum and minimum values for the function.By the way, the answer given is is the minimum and is the maximum value.