$s = \left\{ {0 \in \left( {0,{\pi \over 2}} \right):\sum\limits_{m = 1}^9 {\sec \left( {\theta + (m - 1){\pi \over 6}} \right)\sec \left( {\theta + {{m\pi } \over 6}} \right) = - {8 \over {\sqrt 3 }}} } \right\}$. $\sum\limits_{m = 1}^9 {{1 \over {\cos \left( {\theta + (m - 1){\pi \over 6}} \right)}}\cos \left( {\theta + m{\pi \over 6}} \right)}$ ${1 \over {\sin \left( {{\pi \over 6}} \right)}}\sum\limits_{m = 1}^9 {{{\sin \left[ {\left( {\theta + {{m\pi } \over 6}} \right) - \left( {\theta + (m - 1){\pi \over 6}} \right)} \right]} \over {\cos \left( {\theta + (m - 1){\pi \over 6}} \right)\cos \left( {\theta + m{\pi \over 6}} \right)}}}$ $= 2\sum\limits_{m = 1}^9 {\left[ {\tan \left( {\theta + {{m\pi } \over 6}} \right) - \tan \left( {\theta + (m - 1){\pi \over 6}} \right)} \right]}$ Now, \matrix{ {m = 1} & {2\left[ {\tan \left( {\theta + {\pi \over 6}} \right) - \tan (\theta )} \right]} \cr {m = 2} & {2\left[ {\tan \left( {\theta + {{2\pi } \over 6}} \right) - \tan \left( {\theta + {\pi \over 6}} \right)} \right]} \cr {\matrix{ . \cr . \cr . \cr } } & {} \cr {m = 9} & {2\left[ {\tan \left( {\theta + {{9\pi } \over 6}} \right) - \tan \left( {\theta + 8{\pi \over 6}} \right)} \right]} \cr } $\therefore$ $= 2\left[ {\tan \left( {\theta + {{3\pi } \over 2}} \right) - \tan \theta } \right] = {{ - 8} \over {\sqrt 3 }}$ $= - 2[\cot \theta + \tan \theta ] = {{ - 8} \over {\sqrt 3 }}$ $= - {{2 \times 2} \over {2\sin \theta \cos \theta }} = {{ - 8} \over {\sqrt 3 }}$ $= {1 \over {\sin 2\theta }} = {2 \over {\sqrt 3 }}$ $\Rightarrow \sin 2\theta = {{\sqrt 3 } \over 2}$ $2\theta = {\pi \over 3}$ $2\theta = {{2\pi } \over 3}$ $\theta = {\pi \over 6}$ $\theta = {\pi \over 3}$ $\sum {{\theta _i} = {\pi \over 6} + {\pi \over 3} = {\pi \over 2}}$