Question

If $A,B$ and $C$ are such that $A+B+C=0$, then the value of $\left|\begin{array}{ccc}1& \cos C& \cos B\\ \cos C& 1& \cos A\\ \cos B& \cos A& 1\end{array}\right|$ is

• A. 1
• B. $1-{\cos }^{2}A-{\cos }^{2}B-{\cos }^{2}C-2\cos A\cos B\cos C$
• C. 0
• D. -1

Answer 1

Answer: (C)

$C_2\to C_2-(\cos C)C_1$
$C_3\to C_3-(\cos B)C_1$
$=\left|\begin{array}{ccc}1& 0& 0\\ \cos C& {\sin }^{2}C& \cos A-\cos B\cos C\\ \cos B& \cos A-\cos B\cos C& {\sin }^{2}B\end{array}\right|$
$\because A+B+C=0\Rightarrow A=-(B+C)\Rightarrow \cos A-\cos B\cos C=-\sin B\sin C$
$=\left|\begin{array}{ccc}1& 0& 0\\ \cos C& {\sin }^{2}C& -\sin B\sin C\\ \cos B& -\sin B\sin C& {\sin }^{2}B\end{array}\right|=0$