Question
Evaluate: $\left| {\begin{array}{*{20}{c}} {\cos \alpha \cos \beta }&{\cos \alpha \sin \beta }&{ - \sin \alpha } \\ { - \sin \beta }&{\cos \beta }&0 \\ {\sin \alpha \cos \beta }&{\sin \alpha \sin \beta }&{\cos \alpha } \end{array}} \right|$
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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