Evaluate the determinant: left| begin{array}{ | vedclass

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(A) Let $\Delta = \left| \begin{array}{ccc} \sin^2 x & \cos^2 x & 1 \ \cos^2 x & \sin^2 x & 1 \ -10 & 12 & 2 \end{array} ight|$.Applying the colum

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(A) Let $\Delta = \left| \begin{array}{ccc} \sin^2 x & \cos^2 x & 1 \ \cos^2 x & \sin^2 x & 1 \ -10 & 12 & 2 \end{array} ight|$.Applying the column operation $C_1 	o C_1 + C_2$:$\Delta = \left| \begin{array}{ccc} \sin^2 x + \cos^2 x & \cos^2 x & 1 \ \cos^2 x + \sin^2 x & \sin^2 x & 1 \ -10 + 12 & 12 & 2 \end{array} ight|$Since $\sin^2 x + \cos^2 x = 1$,we have:$\Delta = \left| \begin{array}{ccc} 1 & \cos^2 x & 1 \ 1 & \sin^2 x & 1 \ 2 & 12 & 2 \end{array} ight|$In this determinant,column $C_1$ and column $C_3$ are identical (both are $\begin{bmatrix} 1 \ 1 \ 2 \end{bmatrix}$).According to the properties of determinants,if any two columns are identical,the value of the determinant is $0$.Therefore,$\Delta = 0$.

Evaluate the determinant: $\left| \begin{array}{ccc} \sin^2 x & \cos^2 x & 1 \\ \cos^2 x & \sin^2 x & 1 \\ -10 & 12 & 2 \end{array} \right|$

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