Evaluate | ccc & csin & - - & & 0 & & | - Vidyadip

MCQ

Evaluate $\left|\begin{array}{ccc}\cos \alpha \cos \beta & \cos \alpha \operatorname{csin} \beta & -\sin \alpha \\ -\sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{array}\right|$

A
$2$
B
$-1$
✓
$1$
D
$0$

✓

Answer

Correct option: C.

$1$

c
$\Delta=\left|\begin{array}{ccc}\cos \alpha \cos \beta & \cos \alpha \operatorname{csin} \beta & -\sin \alpha \\ -\sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{array}\right|$

Expanding along $C_{3},$ we have:

$\Delta=-\sin \alpha\left(-\sin \alpha \sin ^{2} \beta+\cos ^{2} \beta \sin a\right)+\cos \alpha\left(\cos \alpha \cos ^{2} \beta+\cos \alpha \sin ^{2} \beta\right)$

$=\sin ^{2} \alpha\left(\sin ^{2} \beta+\cos ^{2} \beta\right)+\cos ^{2} \alpha\left(\cos ^{2} \beta+\sin ^{2} \beta\right)$

$=\sin ^{2} \alpha(1)+\cos ^{2} \alpha(1)$

$=1$

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