Question

The value of the determinant, $\left|\begin{array}{ccc}2\sin A\cos A& \sin (A+B)& \sin (A+C)\\ \sin (A+B)& 2\sin B\cos B& \sin (B+C)\\ \sin (A+C)& \sin (B+C)& 2\sin C\cos C\end{array}\right|$ is:

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is D $0$

The given determinant, $\left|\begin{array}{ccc}2\sin A\cos A& \sin (A+B)& \sin (A+C)\\ \sin (A+B)& 2\sin B\cos B& \sin (B+C)\\ \sin (A+C)& \sin (B+C)& 2\sin C\cos C\end{array}\right|$ can be written as:
$\left|\begin{array}{ccc}\sin A\cos A+\sin A\cos A& \sin A\cos B+\sin B\cos A& \sin A\cos C+\sin C\cos A\\ \sin A\cos B+\sin B\cos A& \sin B\cos B+\sin B\cos B& \sin B\cos C+\sin C\cos B\\ \sin A\cos C+\sin C\cos A& \sin B\cos C+\sin C\cos B& \sin C\cos C+\sin C\cos C\end{array}\right|$
$=\left|\begin{array}{ccc}\sin A& \cos A& 0\\ \sin B& \cos B& 0\\ \sin C& \cos C& 0\end{array}\right|\left|\begin{array}{ccc}\cos A& \sin A& 0\\ \cos B& \sin B& 0\\ \cos C& \sin C& 0\end{array}\right|=0$