Question

Evaluate $\left|\begin{array}{ccc}cos\alpha cos\beta & cos\alpha sin\beta & -sin\alpha \\ -sin\beta & cos\beta & 0\\ sin\alpha cos\beta & sin\alpha sin\beta & cos\alpha \end{array}\right|$

Answer 1

Given, determinant $=\left|\begin{array}{ccc}cos\alpha cos\beta & cos\alpha sin\beta & -sin\alpha \\ -sin\beta & cos\beta & 0\\ sin\alpha cos\beta & sin\alpha sin\beta & cos\alpha \end{array}\right|$
Expanding corresponding to $R_1$, we get
$=cos\alpha cos\beta (cos\alpha cos\beta -0)-cos\alpha sin\beta (-cos\alpha sin\beta -0)-sin\alpha (-si{n}^2\beta sin\alpha -co{s}^2\beta sin\alpha )=co{s}^2\alpha co{s}^2\beta +co{s}^2\alpha si{n}^2\beta +si{n}^2\alpha (si{n}^2\beta +co{s}^2\beta )=co{s}^2\alpha (co{s}^2\beta +si{n}^2\beta )+si{n}^2\alpha (si{n}^2\beta +co{s}^2\beta )=co{s}^2\alpha (co{s}^2\beta +si{n}^2\beta )+si{n}^2\alpha (si{n}^2\beta +co{s}^2\beta )=co{s}^2\alpha \left(1\right)+si{n}^2\alpha \left(1\right)=co{s}^2\alpha +si{n}^2\alpha =1[\because si{n}^2\theta +co{s}^2\theta =1]$