Value of the determinant $| \begin{matrix} \sin 2010 π & \sin \frac{π}{5} \sin \frac{2 π}{5} & \tan \frac{667 π}{4} \\ \sin \frac{3 π}{5} & \cos 2009 \frac{π}{2} & \cos 2013 \frac{π}{2} \\ \tan \frac{3 π}{10} & \cot 2011 \frac{π}{2} & \sin \frac{4 π}{5} \end{matrix} |$ is

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$\frac{5}{8}$

$\frac{- 5}{16}$

$1$

answer is C.

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Detailed Solution

Determinant $= | \begin{matrix} 0 & \sin \frac{π}{5} \sin \frac{2 π}{5} & \tan \frac{667 π}{4} \\ \sin \frac{3 π}{5} & 0 & 0 \\ \tan \frac{3 π}{10} & 0 & \sin \frac{4 π}{5} \end{matrix} |$
$= - \sin \frac{π}{5} \sin \frac{2 π}{5} \sin \frac{3 π}{5} \sin \frac{4 π}{5} = - \sin \frac{π}{5} \sin \frac{2 π}{5} \sin (π - \frac{2 π}{5}) \sin (π - \frac{π}{5}) = - \frac{1}{4} {[\cos \frac{π}{5} - \cos \frac{3 π}{5}]}^{2} = - \frac{1}{4} {[\frac{\sqrt{5} + 1}{4} + \frac{\sqrt{5} - 1}{4}]}^{2} = - \frac{1}{4} \times \frac{5}{4} = \frac{- 5}{16}$