The problem requires us to find the derivative of the function F(x) = cos(x) / sin²(x).
The approach to solve this is to use the quotient rule for derivatives, which states that for functions in the form f(x) = (g(x) / h(x)); the derivative is given by f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x)²).
Applying the quotient rule to this problem, we get:
1. Differentiate the numerator (cos(x)) to get -sin(x).
2. The derivative of the denominator (sin²(x)) is calculated using the chain rule which states that derivative of [f(g(x))] is f'(g(x)).g'(x). The chain rule applied here will give us 2sin(x)cos(x) as the derivative.
3. Apply the quotient rule to get the derivative of the function. Which is = (-sin(x)*sin²(x) - cos(x)*2sin(x)*cos(x)) / (sin²(x))².
Simplifying this expression we get the derivative as: -1/sin(x) - 2cos²(x)/sin³(x).