Official solution
cos210\^{\circ} -- cos10\^{\circ}cos50\^{\circ} + cos250\^{\circ}
= \[{1 \over 2}\][ 2cos210\^{\circ} -- 2cos10\^{\circ}cos50\^{\circ} + 2cos250\^{\circ}]
= \[{1 \over 2}\][ 1 + cos20\^{\circ} - cos60\^{\circ} - cos40\^{\circ} + 1 + cos100\^{\circ}]
= \[{1 \over 2}\][ 2 - \[{1 \over 2}\] + cos20\^{\circ} + cos100\^{\circ} - cos40\^{\circ}]
= \[{1 \over 2}\][ \[{3 \over 2}\] + 2cos60\^{\circ}cos40\^{\circ} - cos40\^{\circ}]
= \[{1 \over 2}\][ \[{3 \over 2}\] + cos40\^{\circ} - cos40\^{\circ}]
= \[{3 \over 4}\]