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Question

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


Solution

$$\left| \begin{matrix} x & \sin { \theta  }  & \cos { \theta  }  \\ -\sin { \theta  }  & -x & 1 \\ \cos { \theta  }  & 1 & x \end{matrix} \right| \\ x({ -x }^{ 2 }-1)-\sin { (-x\sin { \theta  } -\cos { \theta  } ) } +\cos { \theta  } (-\sin { \theta  } +x\cos { \theta  } )\\ =-{ x }^{ 3 }-x+x\sin { ^{ 2 }\theta  } +\sin { \theta  } \cos { \theta  } -\sin { \theta  } \cos { \theta  } +x\cos { ^{ 2 }\theta  } \\ =-{ x }^{ 3 }-x+x\\ =-{ x }^{ 3 }$$
$$ \therefore { -x }^{ 3 }$$ is independent of $$\theta$$.

Mathematics

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