# Solve the following equations

Question:

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

Solution:

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

$\Rightarrow \Delta=(-1)^{1+1} \cos \alpha \cos \beta(\cos \alpha \cos \beta-0)+(-1)^{1+2} \cos \alpha \sin \beta(-\sin \beta \cos \alpha-0)+(-1)^{1+3}(-\sin \alpha)\left(-\sin ^{2} \beta \sin \alpha-\sin \alpha \cos ^{2} \beta\right)$

[Expanding along $R_{1}$ ]

$=\cos \alpha \cos \beta(\cos \alpha \cos \beta-0)-\cos \alpha \sin \beta(-\sin \beta \cos \alpha-0)-\sin \alpha\left(-\sin ^{2} \beta \sin \alpha-\sin \alpha \cos ^{2} \beta\right)$

$=\cos ^{2} \alpha \cos ^{2} \beta+\cos ^{2} \alpha \sin ^{2} \beta+\sin ^{2} \alpha \sin ^{2} \beta+\sin ^{2} \alpha \cos ^{2} \beta$

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

$\Rightarrow \Delta=\cos ^{2} \alpha+\sin ^{2} \alpha \quad\left[\because \sin ^{2} \theta+\cos ^{2} \theta=1\right]$

$\Rightarrow \Delta=1$                                                            $\left[\because \sin ^{2} \theta+\cos ^{2} \theta=1\right]$