#### Please solve RD Sharma class 12 chapter 5 Determinants exercise Fill in the blanks question 33 maths textbook solution

Hint: Here, we use basic concept of determinant of matrix

Given: $D=\left[\begin{array}{ccc} \sin ^{2} A & \cot A & 1 \\ \sin ^{2} B & \cot B & 1 \\ \sin ^{2} C & \cot C & 1 \end{array}\right]$

Solution: Let’s perform some row operations

\begin{aligned} &R_{1} \rightarrow R_{1}-R_{3} \\ &R_{2} \rightarrow R_{2}-R_{3} \\ &{\left[\begin{array}{ccc} \sin ^{2} A-\sin ^{2} C & \cot A-\cot c & 0 \\ \sin ^{2} B-\sin ^{2} C & \cot B-\cot c & 0 \\ \sin ^{2} C & \cot C & 1 \end{array}\right]} \end{aligned}

$=\sin (A-C) \times \sin (B-C)\left[\begin{array}{ccc} \sin B & \frac{1}{\sin A \sin C} & 0 \\ \sin A & \frac{1}{\sin B \sin C} & 0 \\ \sin ^{2} C & \cot c & 1 \end{array}\right]$

\begin{aligned} &=\sin (A-C) \sin (B-C)\left[\frac{1}{\sin C}-\frac{1}{\sin C}\right] \\ &=\sin (A-C) \sin (B-C) \times 0 \\ &=0 \end{aligned}

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