Evaluate $\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \ -sin\beta&cos\beta &0 \ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}$

Question

Evaluate  $\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta  &-sin\alpha \   -sin\beta&cos\beta  &0 \   sin\alpha cos\beta&sin\alpha\sin\beta  &cos\alpha  \end{vmatrix}$

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$\Delta = \begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta  &-sin\alpha \   -sin\beta&cos\beta  &0 \   sin\alpha cos\beta&sin\alpha\sin\beta  &cos\alpha  \end{vmatrix}$ Expanding along C 3 , we have:  Δ=-sinα(-sinαsin 2 β-cos 2 βsinα)+cosα(cosαcos 2 β+cosαsin 2 β) =sin 2 α(sin 2 β+cos 2 β)+cos 2 α(cos 2 β+sin 2 β) =sin 2 α(1)+cos 2 α(1) =1

Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}\)

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