Question

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

• A. $\sin \alpha .\sin \beta .\sin \delta$
• B. $\cos \alpha .\cos \beta .\cos \delta$
• C. $1$
• D. $0$

Answer 1

The correct option is D

$0$

Explanation for correct option:
Given $\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin (\alpha +\gamma )\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin (\beta +\gamma )\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin (\gamma +\delta )\end{array}\right|$
$\Rightarrow \left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin (\alpha +\gamma )\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin (\beta +\gamma )\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin (\gamma +\delta )\end{array}\right|=\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\alpha \right)\cos \left(\gamma \right)+\cos \left(\alpha \right)\sin \left(\gamma \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\beta \right)\cos \left(\gamma \right)+\cos \left(\beta \right)\sin \left(\gamma \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\gamma \right)\cos \left(\delta \right)+\cos \left(\gamma \right)\sin \left(\delta \right)\end{array}\right|\mathbf{}\mathbf{\left[}\mathbf{\because }\mathbf{sin}\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}\mathbf{=}\mathbf{sin}\left(a\right)\mathbf{cos}\left(b\right)\mathbf{+}\mathbf{cos}\left(a\right)\mathbf{sin}\left(b\right)\mathbf{\right]}$
$$\begin{gathered} =\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\alpha \right)\cos \left(\gamma \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\beta \right)\cos \left(\gamma \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \cos \left(\gamma \right)\sin \left(\delta \right)\end{array}\right|+\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \cos \left(\alpha \right)\sin \left(\gamma \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \cos \left(\beta \right)\sin \left(\gamma \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\gamma \right)\cos \left(\delta \right)\end{array}\right| \\ =\cos \left(\gamma \right)\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \sin \left(\alpha \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \sin \left(\beta \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \sin \left(\delta \right)\end{array}\right|+\sin \left(\gamma \right)\left|\begin{array}{ccc}\sin \left(\alpha \right)& \cos \left(\alpha \right)& \cos \left(\alpha \right)\\ \sin \left(\beta \right)& \cos \left(\beta \right)& \cos \left(\beta \right)\\ \sin \left(\delta \right)& \cos \left(\delta \right)& \cos \left(\delta \right)\end{array}\right| \end{gathered}$$
$=\cos \left(\gamma \right)\times 0+\sin \left(\gamma \right)\times 0$ [if two columns are same then the determinant is equal to zero]
$=0$

Hence option(D) is correct