Question

Prove that $\left|\begin{array}{ccc}x& sin\theta & cos\theta \\ -sin\theta & -x& 1\\ cos\theta & 1& x\end{array}\right|$ is independent of $\theta$.

Answer 1

$\left|\begin{array}{ccc}x& \sin \theta & \cos \theta \\ -\sin \theta & -x& 1\\ \cos \theta & 1& x\end{array}\right|x({-x}^2-1)-\sin (-x\sin \theta -\cos \theta )+\cos \theta (-\sin \theta +x\cos \theta )=-x^3-x+x\sin {}^2\theta +\sin \theta \cos \theta -\sin \theta \cos \theta +x\cos {}^2\theta =-x^3-x+x=-x^3$

$\therefore {-x}^3$ is independent of $\theta$.