Question

The value of $\left|\begin{array}{ccc}1& cos(\beta -\alpha )& cos(\gamma -\alpha )\\ cos(\alpha -\beta )& 1& cos(\gamma -\beta )\\ cos(\alpha -\gamma )& cos(\beta -\gamma )& 1\end{array}\right|$ is

• A. ${\left|\begin{array}{ccc}cos\alpha & sin\alpha & 1\\ cos\beta & sin\beta & 1\\ cos\gamma & sin\gamma & 1\end{array}\right|}^2$
• B. ${\left|\begin{array}{ccc}sin\alpha & cos\alpha & 0\\ sin\beta & cos\beta & 0\\ sin\gamma & cos\gamma & 0\end{array}\right|}^2$
• C. ${\left|\begin{array}{ccc}cos\alpha & sin\alpha & 0\\ sin\beta & 0& cos\beta \\ 0& cos\gamma & sin\gamma \end{array}\right|}^2$
• D. $Noneofthese$

Answer 1

The correct option is B ${\left|\begin{array}{ccc}sin\alpha & cos\alpha & 0\\ sin\beta & cos\beta & 0\\ sin\gamma & cos\gamma & 0\end{array}\right|}^2$

$\left|\begin{array}{ccc}1& cos(\beta -\alpha )& cos(\gamma -\alpha )\\ cos(\alpha -\beta )& 1& cos(\gamma -\beta )\\ cos(\alpha -\gamma )& cos(\beta -\gamma )& 1\end{array}\right|-\left|\begin{array}{ccc}co{s}^2\alpha +si{n}^2\alpha & cos\beta cos\alpha +sin\beta sin\alpha & cos\alpha cos\gamma +sin\alpha sin\gamma \\ cos\alpha cos\beta +sin\alpha sin\beta & co{s}^2\beta +si{n}^2\beta & cos\beta cos\gamma +sin\beta sin\gamma \\ cos\alpha cos\gamma +sin\alpha sin\gamma & cos\beta cos\gamma +sin\beta sin\gamma & co{s}^2\beta +si{n}^2\beta \end{array}\right|=\left|\begin{array}{ccc}cos\alpha & sin\alpha & 0\\ cos\beta & sin\beta & 0\\ cos\gamma & sin\gamma & 0\end{array}\right|\left|\begin{array}{ccc}cos\alpha & sin\alpha & 0\\ cos\beta & sin\beta & 0\\ cos\gamma & sin\gamma & 0\end{array}\right|={\left|\begin{array}{ccc}sin\alpha & cos\alpha & 0\\ sin\beta & cos\beta & 0\\ sin\gamma & cos\gamma & 0\end{array}\right|}^2$.