Practice: Finding the sum of several cosines

Geometry Level 3

What value of NN in the interval [0,90] [0, 90] satisfies the following equation: sinN2sin512=cos10+cos20+cos30+cos40+cos50+cos60+cos70+cos80? \begin{aligned} \frac { \sin N ^ \circ } { 2 \sin 5 ^ \circ } - \frac {1}{2} = & \cos 10 ^ \circ + \cos 20 ^ \circ + \cos 30^ \circ + \cos 40 ^ \circ \\ &+ \cos 50 ^ \circ + \cos 60^ \circ + \cos 70 ^ \circ + \cos 80 ^ \circ\, ?\\ \end{aligned}

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