Question

If $2 \sin^{-1} x = \sin^{-1} k$ then $k =$

• A. $2x\sqrt{1-x^{2}}$
• B. $2x$
• C. $x^{2}$
• D. $x\sqrt{1-2x^{2}}$

Hint:

Inverse trig formulas often mirror their standard trig counterparts. $2\theta \to 2\sin\theta\cos\theta$.

Answer 1

The Correct Option is A

Step 1: Concept
This is the standard formula for the double angle of the inverse sine function.

Step 2: Meaning
Let $\sin^{-1} x = \theta$, then $x = \sin \theta$. The equation becomes $2\theta = \sin^{-1} k$, which means $k = \sin(2\theta)$.

Step 3: Analysis
Using the identity $\sin 2\theta = 2 \sin \theta \cos \theta$. Since $\sin \theta = x$, then $\cos \theta = \sqrt{1 - x^2}$. Thus, $k = 2x\sqrt{1 - x^2}$.

Step 4: Conclusion
Comparing with the options, $k$ corresponds to $2x\sqrt{1-x^{2}}$.
Final Answer: (A)