Question

\( \cos\theta (\csc\theta - \sec\theta) - \cot\theta = \)

• A. -1
• B. 1
• C. 0
• D. \( \cos^2\theta - \tan^2\theta \)

Hint:

When simplifying complex trigonometric expressions, it's often helpful to convert all terms into sines and cosines.
Use fundamental identities: \(\csc\theta = 1/\sin\theta\), \(\sec\theta = 1/\cos\theta\), \(\cot\theta = \cos\theta/\sin\theta\).
Look for terms that cancel out or simplify.

Answer 1

The Correct Option is A

Let the expression be E. \( E = \cos\theta (\csc\theta - \sec\theta) - \cot\theta \) Convert all terms to \(\sin\theta\) and \(\cos\theta\): \(\csc\theta = \frac{1}{\sin\theta}\) \(\sec\theta = \frac{1}{\cos\theta}\) \(\cot\theta = \frac{\cos\theta}{\sin\theta}\) Substitute these into the expression: \( E = \cos\theta \left( \frac{1}{\sin\theta} - \frac{1}{\cos\theta} \right) - \frac{\cos\theta}{\sin\theta} \) Distribute \(\cos\theta\) in the first term: \( E = \cos\theta \cdot \frac{1}{\sin\theta} - \cos\theta \cdot \frac{1}{\cos\theta} - \frac{\cos\theta}{\sin\theta} \) \( E = \frac{\cos\theta}{\sin\theta} - 1 - \frac{\cos\theta}{\sin\theta} \) The terms \(\frac{\cos\theta}{\sin\theta}\) and \(-\frac{\cos\theta}{\sin\theta}\) cancel out. \( E = -1 \) This matches option (a). \[ \boxed{-1} \]