Question:
Evaluate
\[
\cos\theta(\cosec\theta-\sec\theta)-\cot\theta
\]
Evaluate
\[
\cos\theta(\cosec\theta-\sec\theta)-\cot\theta
\]
Show Hint
In trigonometric simplification problems, first convert all functions into \(\sin\theta\) and \(\cos\theta\). This often leads to immediate cancellation of terms.
Updated On: Jun 26, 2026
- \(-1\)
- \(1\)
- \(0\)
- \(\cos^2\theta-\tan^2\theta\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Replace \(\cosec\theta\) and \(\sec\theta\) by their definitions.
Using \[ \cosec\theta=\frac{1}{\sin\theta} \] and \[ \sec\theta=\frac{1}{\cos\theta} \] the given expression becomes \[ \cos\theta \left( \frac{1}{\sin\theta} - \frac{1}{\cos\theta} \right) -\cot\theta \]
Step 2: Multiply \(\cos\theta\) inside the bracket.
\[ = \frac{\cos\theta}{\sin\theta} -\frac{\cos\theta}{\cos\theta} -\cot\theta \] \[ = \cot\theta-1-\cot\theta \]
Step 3: Simplify.
Cancelling \(\cot\theta\), \[ = -1 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{-1} \] Hence, the correct option is \[ \boxed{(1)\ -1} \]
Using \[ \cosec\theta=\frac{1}{\sin\theta} \] and \[ \sec\theta=\frac{1}{\cos\theta} \] the given expression becomes \[ \cos\theta \left( \frac{1}{\sin\theta} - \frac{1}{\cos\theta} \right) -\cot\theta \]
Step 2: Multiply \(\cos\theta\) inside the bracket.
\[ = \frac{\cos\theta}{\sin\theta} -\frac{\cos\theta}{\cos\theta} -\cot\theta \] \[ = \cot\theta-1-\cot\theta \]
Step 3: Simplify.
Cancelling \(\cot\theta\), \[ = -1 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{-1} \] Hence, the correct option is \[ \boxed{(1)\ -1} \]
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