Function of Birmingham

A Birmingham city representative proposed an amazing problem to the king of Sussex, claiming a deal of a million dollars.The King being cool himself reduced it to this and minimal constant value which is positive is the answer to the representative's question.

1] 8f(x)f(y) = (( cos(yx)+i sin(yx) )^(1/y)) (cos(x) - i sin(x)) + (( cos(yx) + i sin(yx) ) ^(1/x)) (cos(y)- i sin(y)) .

2] f(x) + f(y) = (cos(x^2 y^2) + i sin(x^2*y^2))^ (2/xy) - ( sin(2xy) - 2(sin(xy)^(2)) ). Therefore, find the total number of positive solutions =x, the sum of all such solutions = y and the smallest solution z, and evaluate 1000x +9870y + 53700z.