A certain joint probability density function is given by the formula:

\[f_{XY} (xy) = \dfrac{\sqrt{\pi}}{2} x\sin (xy),\]

where \(x\) and \(y\) are drawn from the rectangle \([0,\sqrt{\pi}]\times [0,\sqrt{\pi}]\).

Are the random variables \(X\) and \(Y\) independent?

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