# 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 $$\alpha + \beta$$.

×