Question

The determinant $\left|\begin{array}{ccc}cos(\theta +\varphi )& -sin(\theta +\varphi )& cos2\varphi \\ sin\theta & cos\theta & sin\varphi \\ -cos\theta & sin\theta & cos\varphi \end{array}\right|$ is

• A. 0
• B. independent of $\theta$
• C. independent of $\varphi$
• D. independent of both $\theta$ and $\varphi$

Answer 1

The correct option is B independent of $\theta$

$\left|\begin{array}{ccc}\cos (\theta +\varphi )& -\sin (\theta +\varphi )& \cos 2\varphi \\ \sin \theta & \cos \theta & \sin \varphi \\ -\cos \theta & \sin \theta & \cos \varphi \end{array}\right|$

$=\cos (\theta +\varphi )\left(\cos \theta \cos \varphi -\sin \theta \sin \varphi \right)+\sin (\theta +\varphi )\left(\sin \theta \cos \varphi +\cos \theta \sin \varphi \right)+\cos 2\varphi \left({\sin }^2\theta +{\cos }^2\theta \right)$

$={\cos }^2(\theta +\varphi )+{\sin }^2(\theta +\varphi )+\cos 2\varphi =1+\cos 2\varphi$