Question

If $\alpha ,\beta ,\gamma$ are real numbers then$△=\left|\begin{array}{ccc}1& \cos (\beta -\alpha )& \cos (\gamma -\alpha )\\ \cos (\alpha -\beta )& 1& \cos (\gamma -\beta )\\ \cos (\alpha -\gamma )& \cos (\beta -\gamma )& 1\end{array}\right|$ is equal to

• A. $-1$
• B. $\cos \alpha \cos \beta \beta \cos \gamma$
• C. $\cos \alpha +\cos \beta +\cos \gamma$
• D. $0$

Answer 1

The correct option is D $0$

$A=\left[\begin{array}{ccc}1& \cos (\beta -\alpha )& \cos (\gamma -\alpha )\\ \cos (\alpha -\beta )& 1& \cos (\gamma -\beta )\\ \cos (\alpha -\gamma )& \cos (\beta -\gamma )& 1\end{array}\right]$

$A=\left[\begin{array}{ccc}1& \cos \beta \cos \alpha +\sin \beta \sin \alpha & \cos \gamma \cos \alpha +\sin \gamma \sin \alpha \\ \cos \alpha \cos \beta +\sin \alpha \sin \beta & 1& \cos \gamma \cos \beta +\sin \gamma \sin \beta \\ \cos \alpha \cos \beta +\sin \alpha \sin \beta & \cos \beta \cos \gamma +\sin \beta \sin \gamma & 1\end{array}\right]$

$A=\left[\begin{array}{ccc}\cos \alpha & sin\alpha & 0\\ \cos \beta & \sin \beta & 0\\ \cos \gamma & \sin \beta & 0\end{array}\right]\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0\\ \cos \beta & \sin \beta & 0\\ \cos \gamma & \sin \gamma & 0\end{array}\right]$

$A=0$.