Question

The determinant $\left|\begin{array}{ccc}\sin A& \cos A& \sin A+\cos B\\ \sin B& \cos A& \sin B+\cos B\\ \sin C& \cos A& \sin C+\cos B\end{array}\right|$ is equal to

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

Answer 1

The correct option is A $0$

$\left|\begin{array}{ccc}\sin A& \cos A& \sin A+\cos B\\ \sin B& \cos A& \sin B+\cos B\\ \sin C& \cos A& \sin C+\cos B\end{array}\right|=\left|\begin{array}{ccc}\sin A& \cos A& \sin A\\ \sin B& \cos A& \sin B\\ \sin C& \cos A& \sin C\end{array}\right|+\left|\begin{array}{ccc}\sin A& \cos A& \cos B\\ \sin B& \cos A& \cos B\\ \sin C& \cos A& \cos B\end{array}\right|\text{Taking}\cos A,\cos B\text{common from}{C}_2,C_3\text{respectively from second determinant}=0+\cos A\cos B\left|\begin{array}{ccc}\sin A& 1& 1\\ \sin B& 1& 1\\ \sin C& 1& 1\end{array}\right|\{\because C_1,C_3\text{and}{C}_2,C_3\text{are same in first and second determinants respectively}\}=0+0$