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