Explanation

(c)
$\left| \begin{matrix}
cos(x - a) & cos(x + a) & \cos x \\
\sin{(x + a)} & sin(x - a) & \sin x \\
{cos\ a}{{\ tan}x} & \cos a\cot x & cosec\ 2x
\end{matrix} \right|$
$\left| \begin{matrix}
\cos(x - a) + cos(x - a) & cos(x + a) & \cos x \\
\sin(x + a) + sin(x - a) & sin(x - a) & \sin x \\
{cos\ a}{{\ (\tan}{x + \cot x)}} & \cos a\cot x & cosec\ 2x
\end{matrix} \right|$
$= \left| \begin{matrix}
2\cos{x\cos a} & cos(x + a) & \cos x \\
2\sin{x\cos a} & sin(x - a) & \sin x \\
{cos\ a}\left( \frac{\tan^{2}{x + 1}}{\tan x} \right) & \cos a\cot x & cosec\ 2x
\end{matrix} \right|$
$= 2\cos{a\ \left| \begin{matrix}
cosx & cos(x + a) & \cos x \\
\sin x & sin(x - a) & \sin x \\
cosec\ 2x & \cos a\cot x & cosec\ 2x
\end{matrix} \right|} = 0$
[$\because$ two columns are identical]