Question
Evaluate $\triangle=\begin{vmatrix}0&\sin\alpha&-\cos\alpha\\-\sin\alpha&0&\sin\beta\\\cos\alpha&-\sin\beta&0 \end{vmatrix}$
✓
Answer
Let $\triangle=\begin{vmatrix}0&\sin\alpha&-\cos\alpha\\-\sin\alpha&0&\sin\beta\\\cos\alpha&-\sin\beta&0 \end{vmatrix}$
$\triangle=(-1)^{1+1}0(0+\sin^2\beta)+(-1)^{1+2}\sin\alpha(0-\sin\beta\cos)\beta\\+(-1)^{1+3}(-\cos\alpha)(\sin\alpha\sin\beta-0)$ [Expanding along R1]
$=0(0+\sin^2\beta)-\sin\alpha(0-\sin\beta\cos\alpha)-\cos\alpha(\sin\alpha\sin\beta-0)$
$=\sin\alpha\sin\beta\cos\alpha-\sin\alpha\sin\beta\cos\alpha$
$=0$
$\triangle=(-1)^{1+1}0(0+\sin^2\beta)+(-1)^{1+2}\sin\alpha(0-\sin\beta\cos)\beta\\+(-1)^{1+3}(-\cos\alpha)(\sin\alpha\sin\beta-0)$ [Expanding along R1]
$=0(0+\sin^2\beta)-\sin\alpha(0-\sin\beta\cos\alpha)-\cos\alpha(\sin\alpha\sin\beta-0)$
$=\sin\alpha\sin\beta\cos\alpha-\sin\alpha\sin\beta\cos\alpha$
$=0$
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