Question

Evaluate \(\begin{vmatrix}-\sin\theta&\cos\theta \\ \sec\theta&\csc\theta\end{vmatrix}\).

• A. \(0\)
• B. \(-1\)
• C. \(-2\)
• D. \(-\sin 2\theta\)

Hint:

Replace \(\sec,\csc\) by reciprocals and cancel—often the quickest path.

Answer 1

The Correct Option is C

Idea. For a \(2\times2\) matrix \(\begin{vmatrix}a&b \\ c&d\end{vmatrix}=ad-bc\). Use reciprocal trig identities \(\sec\theta=\tfrac1{\cos\theta}\), \(\csc\theta=\tfrac1{\sin\theta}\).
Step. \[ (-\sin\theta)\left(\frac{1}{\sin\theta}\right)-\cos\theta\left(\frac{1}{\cos\theta}\right)=-1-1=-2. \]