Question

# Prove that the determinant $$\begin{vmatrix} x & sin \theta & cos \theta \\ - sin \theta & - x & 1 \\ cos \theta & 1 & x \end{vmatrix}$$ is independent of $$\theta$$.

Solution

## $$\begin{vmatrix} x & sin \theta & cos \theta \\ - sin \theta & - x & 1 \\ cos \theta & 1 & x \end{vmatrix}$$$$= x(-x^{2}-1)- sin \theta (-x sin\theta - cos\theta)+cos \theta (- sin\theta +x cos \theta )$$$$= -x^{3}-x+x sin^{2}\theta + sin\theta cos\theta -sin\theta cos\theta +x cos^{2}\theta$$$$= -x^{3}-x+x(sin^{2}\theta +cos^{2}\theta )$$$$=-x^{3}-x+x$$$$=-x^{3}$$ ( Which is Independent of $$\theta$$ )Mathematics

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