Given that $$-4 \sum_{k=1}^{1009} \frac{\sin (2k^\circ)\sin 1^\circ}{\cos (4k^\circ)+\cos 2^\circ}$$can be written in the form $\dfrac 1{\cos 1^\circ}-\dfrac 1{\cos n^\circ},$ where $n \in \mathbb Z^+,$ find the minimum value of $n$.

Source: Friend of mine

I thought of listing terms out, but that got me nowhere. There were too many terms. Additionally, the solution that he gave was very strange (there were a lot of spaces in his computation). Can someone guide me through the solution?


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