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यदि $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$ ,तब
$A = 0$, $\theta $ के सभी मानों के लिये
$A$, , का एक विषम फलन है
$A = 0$, $\theta = \alpha + \beta + \gamma $ के लिये
$A$, $\theta $ से स्वतंत्र है
Solution
(d) दिया है $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$
संक्रिया ${R_2} \to {R_2} – {R_1},\,{R_3} \to {R_3} – {R_1}$ से,
$\therefore $$A = \{ \cos (\theta + \gamma ) – \cos (\theta + \alpha )\} $
$\{ \sin (\theta + \beta ) – \sin (\theta + \alpha )\} – \{ \cos (\theta + \beta )$
= $\sin (\beta – \gamma ) – \sin (\beta – \alpha ) – \sin (\alpha – \gamma )$,
जो कि $\theta $से स्वतंत्र है।
