Question

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

Answer 1

$\left|\begin{array}{ccc}x& sin\theta & cos\theta \\ -sin\theta & -x& 1\\ cos\theta & 1& x\end{array}\right|$

$=x(-x^2-1)-sin\theta (-xsin\theta -cos\theta )+cos\theta (-sin\theta +xcos\theta )$

$=-x^3-x+xsi{n}^2\theta +sin\theta cos\theta -sin\theta cos\theta +xco{s}^2\theta$

$=-x^3-x+x(si{n}^2\theta +co{s}^2\theta )$

$=-x^3-x+x$

$=-x^3$ ( Which is Independent of $\theta$ )