Let the determinant be $\Delta=\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$

Which of the following statements are correct?

(A) $\Delta$ is independent of $\theta$
(B) $\Delta$ is independent of x
(C) $\Delta$ is independent of $\theta$ and x both and is equal to 1
(D) $\Delta$ depends upon both of x and $\theta$
(E) $\Delta$ is independent of $\theta$ and is equal to $x^3$

Choose the correct answer from the options given below:

(A) Only

The correct answer is Option (1) → (A) Only

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

$=x(-x^2-1)+\sin θ(\cos θ+x\sin θ)+\cos θ(-\sin θ+x\cos θ)$

$=-x^3-x+\sin θ\cos θ+x\sin^2θ-\sin θ\cos θ+x^2\cos^2θ$

$=-x^3-x+x(\sin^2θ+\cos^2θ)$

$⇒-x^3$