Question
Evaluate: $\left|\begin{array}{ccc}\cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin \alpha \\ -\sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{array}\right|$
Expanding along first row,
$= \cos \alpha \cos \beta \left( {\cos \alpha \cos \beta - 0} \right) $ $- \cos \alpha \sin \beta \left( { - \cos \alpha \sin \beta - 0} \right) $ $- \sin \alpha \left( { - \sin \alpha {{\sin }^2}\beta - \sin \alpha {{\cos }^2}\beta } \right)$
$= {\cos ^2}\alpha {\cos ^2}\beta + {\cos ^2}\alpha {\sin ^2}\beta $ $+ {\sin ^2}\alpha \left( {{{\sin }^2}\beta + {{\cos }^2}\beta } \right)$
$= {\cos ^2}\alpha \left( {{{\cos }^2}\beta + {{\sin }^2}\beta } \right)$ $ + {\sin ^2}\alpha \left( {{{\sin }^2}\beta + {{\cos }^2}\beta } \right)$
$= {\cos ^2}\alpha + {\sin ^2}\alpha$
$=$ 1
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$\int\text{e}^{\text{x}}\big[\sec\text{x}+\log(\sec\text{x}+\tan\text{x})\big]\text{dx}$