To find the derivative of F(x) = sin(x)/(1 + csc^2(x)), we can use the quotient rule.

Let's rewrite the function as F(x) = sin(x)/(1 + cot^2(x)).

Now, we can differentiate using the quotient rule:

F'(x) = (1 + cot^2(x))(cos(x)) - sin(x)(2cot(x)(-csc^2(x)) / (1 + cot^2(x))^2

F'(x) = cos(x) + cos(x)cot^2(x) - 2sin(x)cot(x)(-csc^2(x)) / (1 + cot^2(x))^2

F'(x) = cos(x) + cos(x) * cot^2(x) + 2sin(x)cot(x)csc^2(x) / (1 + cot^2(x))^2

Therefore, the derivative of F(x) = sin(x)/(1 + cot^2(x)) is F'(x) = cos(x) + cos(x) * cot^2(x) + 2sin(x)cot(x)csc^2(x) / (1 + cot^2(x))^2.