Question
Evaluate : $\left|\begin{array}{lll}\sec \theta & \tan \theta & 0 \\ \tan \theta & \sec \theta & 0 \\ 0 & 0 & 1\end{array}\right|$
✓
Answer
$\begin{aligned} & \left|\begin{array}{lll}\sec \theta & \tan \theta & 0 \\ \tan \theta & \sec \theta & 0 \\ 0 & 0 & 1\end{array}\right| \\ & =\sec \theta\left|\begin{array}{cc}\sec \theta & 0 \\ 0 & 1\end{array}\right|-\tan \theta\left|\begin{array}{cc}\tan \theta & 0 \\ 0 & 1\end{array}\right| \\ & +0\left|\begin{array}{cc}\tan \theta & \sec \theta \\ 0 & 0\end{array}\right| \\ & =\sec \theta(\sec \theta-0)-\tan \theta(\tan \theta-0)+0 \\ & =\sec ^2 \theta-\tan ^2 \theta \\ & =1 \end{aligned}$
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