Question

If $A, B$ and $C$ are such that $A+B+C=0$, then the value of $\begin{vmatrix}1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1\end{vmatrix}$ 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)

$C _2 \rightarrow C _2-(\cos C ) C _1$
$C _3 \rightarrow C _3-(\cos B ) C _1$
$=\begin{vmatrix}1 & 0 & 0 \\ \cos C & \sin ^2 C & \cos A -\cos B \cos C \\ \cos B & \cos A -\cos B \cos C & \sin ^2 B \end{vmatrix}$
$\because A+B+C=0 \Rightarrow A=-(B+C) \Rightarrow \cos A-\cos B \cos C=-\sin B \sin C$
$=\begin{vmatrix}1 & 0 & 0 \\ \cos C & \sin ^2 C & -\sin B \sin C \\ \cos B & -\sin B \sin C & \sin ^2 B \end{vmatrix}=0$