Without expanding show that: `=|cos e c^2thetacot^2theta1cot^2thetacos e c^2theta-1 42 40 2|=0`

Answer

Proved

Explanation

Applying `C_{1} \rightarrow C_{1}-C_{2}-C_{3}`, we have
$$
\Delta=\left|\begin{array}{ccc}
\operatorname{cosec}^{2} \theta-\cot ^{2} \theta-1 & \cot ^{2} \theta & 1 \\
\cot ^{2} \theta-\operatorname{cosec}^{2} \theta+1 & \operatorname{cosec}^{2} \theta & -1 \\
0 & 40 & 2
\end{array}\right|=\left|\begin{array}{ccc}
0 & \cot ^{2} \theta & 1 \\
...