Question

$2 \tan^-1(\csc \tan^-1x - \tan \cot^-1x)$ is equal to

• A. $\cot^{-1}x$
• B. $\cot^{-1}\frac{1}{x}$
• C. $\tan^{-1}x$
• D. None of these

Hint:

$2 \tanx - \tan \cotx)$ is equal to

Answer 1

The Correct Option is C

Step 1: Concept
Simplify inverse trigonometric terms by using substitutions or identities.
Step 2: Analysis
Let $x = \tan \theta$. Then $\tan^{-1}x = \theta$ and $\cot^{-1}x = \pi/2 - \theta$.
Step 3: Evaluation
The expression inside becomes $\csc \theta - \tan(\pi/2 - \theta) = \csc \theta - \cot \theta = \frac{1-\cos \theta}{\sin \theta} = \tan(\theta/2)$.
Step 4: Conclusion
$2 \tan^{-1}(\tan \frac{\theta}{2}) = 2 \cdot \frac{\theta}{2} = \theta = \tan^{-1}x$.
Final Answer: (c)