Let $x$ be a real number in the interval $\left(0,\dfrac{\pi}{2}\right)$ such that $\dfrac{1}{\sin x \cos x}+2\cot 2x=\dfrac{1}{2}$.  Evaluate $\dfrac{1}{\sin x \cos x}-2\cot 2x$.

Let's think step by step and output the final answer within \boxed{}.
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