Question

Evaluate \[ \cos\theta(\cosec\theta-\sec\theta)-\cot\theta \]

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

Hint:

In trigonometric simplification problems, first convert all functions into \(\sin\theta\) and \(\cos\theta\). This often leads to immediate cancellation of terms.

Answer 1

The Correct Option is A

Step 1: Expand using definitions.
\[\cos\theta(\csc\theta - \sec\theta) - \cot\theta = \cos\theta\cdot\frac{1}{\sin\theta} - \cos\theta\cdot\frac{1}{\cos\theta} - \frac{\cos\theta}{\sin\theta}.\] \[= \frac{\cos\theta}{\sin\theta} - 1 - \frac{\cos\theta}{\sin\theta} = -1.\]
\[\boxed{-1}\]