If $\sec ^2 A \hat{i}+\hat{j}+\hat{k}, \hat{i}+\sec ^2 B \hat{j}+\hat{k}$, and $\hat{i}+\hat{j}+\sec ^2 C \hat{k}$ are coplanar, then $\cot ^2 A+\cot ^2 B+\cot ^2 C$ is

equal to 0

The vectors are co-planar

$\Rightarrow\left|\begin{array}{ccc}
\sec ^2 A & 1 & 1 \\
1 & \sec ^2 B & 1 \\
1 & 1 & \sec ^2 C
\end{array}\right|=0$

$\Rightarrow \cot ^2 A+\cot ^2 B+\cot ^2 C+1=0$ which is not possible.

Hence (3) is correct answer.