Question

The value of the determinant $\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\alpha +\gamma \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\beta +\gamma \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\delta +\gamma \right)\end{array}\right|$ is

• A. $\sin \left(\alpha \right)\sin \left(\beta \right)\sin \left(\delta \right)$
• B. $\cos \left(\alpha \right)\cos \left(\beta \right)\cos \left(\delta \right)$
• C. $1$
• D. $0$

Answer 1

The correct option is D

$0$

Explanation for correct option

$$\begin{gathered} \left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\alpha +\gamma \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\beta +\gamma \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\delta +\gamma \right)\end{array}\right| \\ =\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\alpha \right)\cos \left(\gamma \right)+\sin \left(\gamma \right)\cos \left(\alpha \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\beta \right)\cos \left(\gamma \right)+\sin \left(\gamma \right)\cos \left(\beta \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\delta \right)\cos \left(\gamma \right)+\sin \left(\gamma \right)\cos \left(\delta \right)\end{array}\right|\left[\because \sin \left(A+B\right)=\sin \left(A\right)\cos \left(B\right)+\sin \left(B\right)\cos \left(A\right)\right] \\ =\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\alpha \right)\cos \left(\gamma \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\beta \right)\cos \left(\gamma \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\delta \right)\cos \left(\gamma \right)\end{array}\right|+\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\gamma \right)\cos \left(\alpha \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\gamma \right)\cos \left(\beta \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\gamma \right)\cos \left(\delta \right)\end{array}\right|\left[\because \left|\begin{array}{ccc}a+b& g& j\\ c+d& h& k\\ e+f& i& l\end{array}\right|=\left|\begin{array}{ccc}a& g& j\\ c& h& k\\ e& i& l\end{array}\right|+\left|\begin{array}{ccc}b& g& j\\ d& h& k\\ f& i& l\end{array}\right|\right] \end{gathered}$$

$=\cos \left(\gamma \right)\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\alpha \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\beta \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\delta \right)\end{array}\right|+\sin \left(\gamma \right)\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \cos \left(\alpha \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \cos \left(\beta \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \cos \left(\delta \right)\end{array}\right|$

By taking $\cos \left(\gamma \right)$ from third column of the $1^{\mathrm{st}}$ determinant and $\sin \left(\gamma \right)$ from third column of the $2^{\mathrm{nd}}$ determinant

In $1^{\mathrm{st}}$ determinant $C_1=C_3$ and in $2^{\mathrm{nd}}$ determinant $C_2=C_3$

$\left|A\right|=\cos \left(\gamma \right)\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\alpha \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\beta \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\delta \right)\end{array}\right|+\sin \left(\gamma \right)\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \cos \left(\alpha \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \cos \left(\beta \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \cos \left(\delta \right)\end{array}\right|$

$=\cos \left(\gamma \right)\left(0\right)+\sin \left(\gamma \right)\left(0\right)$ [Since determinant having same rows or column have zero values]

$=0$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{D}\right)$