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सारणिकों के गुणधर्मो का प्रयोग करके निम्नलिखित प्रश्न को सिद्ध कीजिए :
$\left| {\begin{array}{*{20}{l}}
{\sin \alpha }&{\cos \alpha }&{\cos (\alpha + \delta )} \\
{\sin \beta }&{\cos \beta }&{\cos (\beta + \delta )} \\
{\sin \gamma }&{\cos \gamma }&{\cos (\gamma + \delta )}
\end{array}} \right| = 0$
Solution
$\Delta=\left|\begin{array}{lll}\sin \alpha & \cos \alpha & \cos (\alpha+\delta) \\ \sin \beta & \cos \beta & \cos (\beta+\delta) \\ \sin \gamma & \cos \gamma & \cos (\gamma+\delta)\end{array}\right|$
$=\frac{1}{\sin \delta \cos \delta}\left|\begin{array}{ccc}\sin \alpha \sin \delta & \cos \alpha \cos \delta & \cos \alpha \cos \delta-\sin \alpha \sin \delta \\ \sin \beta \sin \delta & \cos \beta \cos \delta & \cos \beta \cos \delta-\sin \beta \sin \delta \\ \sin \gamma \sin \delta & \cos \gamma \cos \delta & \cos \gamma \cos \delta-\sin \gamma \sin \delta\end{array}\right|$
Applying $C_{1} \rightarrow+C_{1}+C_{3},$ we have:
$\Delta=\frac{1}{\sin \delta \cos \delta}\left|\begin{array}{lll}
\cos \alpha \cos \delta & \cos \alpha \cos \delta & \cos \alpha \cos \delta-\sin \alpha \sin \delta \\
\cos \beta \cos \delta & \cos \beta \cos \delta & \cos \beta \cos \delta-\sin \beta \sin \delta \\
\cos \gamma \cos \delta & \cos \gamma \cos \delta & \cos \gamma \cos \delta-\sin \gamma \sin \delta
\end{array}\right|$
Here, two columns $C_{1}$ and $C_{2}$ are identical.
$\therefore \Delta=0$
Hence, the given result is proved.
