Question

The value of \( \sin^{-1}\left(\frac{2x}{1 + x^{2}}\right) \) is:

• A. $2 \tan^{-1} x$
• B. $\tan^{-1} 2x$
• C. $2 \sin^{-1} x$
• D. None of these

Hint:

This substitution is valid for $|x| \leq 1$. For $|x|>1$, the formula changes based on the principal branch.

Answer 1

The Correct Option is A

Step 1: Concept
Use trigonometric substitution to simplify the inverse function.
Step 2: Analysis
Substitute $x = \tan \theta$. Then $\frac{2x}{1+x^{2}} = \frac{2\tan \theta}{1+\tan^{2} \theta} = \sin 2\theta$. The expression becomes $\sin^{-1}(\sin 2\theta) = 2\theta$.
Step 3: Conclusion
Since $\theta = \tan^{-1} x$, the result is $2 \tan^{-1} x$.
Final Answer: (A)