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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 $\theta$.
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 $\theta$.
CBSE | Class 12 | Class 12 | Determinants | Maths | Miscellaneous | NCERT
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 $\theta$.

Let $\Delta=\left[\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right]$

$\Delta=x\left|\begin{array}{cc}-x & 1 \\ 1 & x\end{array}\right|-\sin \theta\left|\begin{array}{cc}-\sin \theta & 1 \\ \cos \theta & x\end{array}\right|+\cos \theta\left|\begin{array}{cc}-\sin \theta & -x \\ \cos \theta & 1\end{array}\right|$

$=x\left(-x^{2}-1\right)-\sin \theta(-x \sin \theta-\cos \theta)+\cos \theta(-\sin \theta+x \cos \theta)$

=-x3-x+xsin2θ+cos2θ
=-x^{3}-x+x\left(\sin ^{2} \theta+\cos ^{2} \theta\right)

=-x3
=-x^{3}

Which is independent of $\theta$

i

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