\[\large \displaystyle\int \dfrac{\sec^2(x) - 2010}{\sin^{2010}(x)}\, dx = \dfrac{P(x)}{(\sin(x))^{2010}} + C \]

If the equation above is true for some function \(P(x) \) with \( P ( \frac{ \pi}{4} ) = \frac{1}{2^{1005} } \), where \(C\) denotes the arbitrary constant, then find the value of \( P\left(\dfrac{\pi}{3}\right)\).

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