Question

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

Answer 1

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

Expanding along C3, we have:
Δ=-sinα(-sinαsin2β-cos2βsinα)+cosα(cosαcos2β+cosαsin2β)
=sin2α(sin2β+cos2β)+cos2α(cos2β+sin2β)
=sin2α(1)+cos2α(1)
=1