Explanation

(a)
$\left| \begin{matrix}
\cos{(A - P)} & \cos{(A - Q)} & \cos{(A - R)} \\
\cos{(B - P)} & \cos{(B - Q)} & \cos{(B - R)} \\
\cos{(C - P)} & \cos{(C - Q)} & \cos{(C - R)}
\end{matrix} \right|$
$= \left| \begin{matrix}
\cos{A\cos{P + \sin{A\sin P}}} & \cos{A\cos{Q + \sin{A\sin Q}}} & \cos{A\cos{R + \sin{A\sin R}}} \\
\cos{B\cos{P + \sin{B\sin P}}} & \cos{B\cos Q} + \sin{B\sin Q} & \cos{B\cos{R + \sin{B\sin R}}} \\
\cos{C\cos{P + \sin{C\sin P}}} & \cos{C\cos Q} + \sin{C\sin Q} & \cos{C\cos{R + \sin{C\sin R}}}
\end{matrix} \right|$
The determinants can be rewritten as 8 determinants and the value of each of these 8 determinants is zero.
$ie,\cos{P\cos{Q\cos{R\left| \begin{matrix}
\cos A & \cos A & \cos A \\
\cos B & \cos B & \cos B \\
\cos C & \cos C & \cos C
\end{matrix} \right| = 0}}}$
Similarly, other determinants can be shown zero.