Evaluate $\Delta=\left|\begin{array}{ccc}0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$
Evaluate $\Delta=\left|\begin{array}{ccc}0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$
Expanding along $\mathrm{R}_{1},$ we get
$\Delta {\text{ }} = 0\left| {\begin{array}{*{20}{c}}
0&{\sin \beta } \\
{ - \sin \beta }&0
\end{array}} \right| - \sin \alpha \left| {\begin{array}{*{20}{c}}
{ - \sin \alpha }&{\sin \beta } \\
{\cos \alpha }&0
\end{array}} \right| - \cos \alpha \left| {\begin{array}{*{20}{c}}
{ - \sin \alpha }&0 \\
{\cos \alpha }&{ - \sin \beta }
\end{array}} \right|$
$=0-\sin \alpha(0-\sin \beta \cos \alpha)-\cos \alpha(\sin \alpha \sin \beta-0)$
$=\sin \alpha \sin \beta \cos \alpha-\cos \alpha \sin \alpha \sin \beta=0$
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- [JEE MAIN 2019]
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