Everything Just Disappear?

Geometry Level 4

\[\large \dfrac{\sin 2^\circ}{\cos(3\times2^\circ)} + \dfrac{\sin(3\times 2^\circ)}{\cos (3^2\times2^\circ) } + \dfrac{\sin(3^2\times 2^\circ)}{\cos (3^3\times2^\circ) } + \cdots + \dfrac{\sin(3^9\times 2^\circ)}{\cos (3^{10}\times2^\circ) } \]

If the expression above can be expressed as \( \dfrac12 (\tan\alpha^\circ - \tan\beta^\circ )\), where \(\alpha\) and \(\beta\) are positive integers, find the minimum value of \(\alpha + \beta\).

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