Show that(i) `sin^(-1)(2xsqrt(1-x^2))=2sin^(-1)x ,-1/(sqrt(2))lt=xlt=1/(sqrt(2))`(ii) `sin^(-1)(2xsqrt(1-x^2))=2cos^(-1)x ,1/(sqrt(2))lt=xlt=1`

Explanation

To solve the given problem, we will break it down into two parts as stated in the question.

### Part (i): Show that \( \sin^{-1}(2x\sqrt{1-x^2}) = 2\sin^{-1}x \) for \( -\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}} \).

**Step 1:** Start with the left-hand side (LHS):
\[ \text{LHS} = \sin^{-1}(2x\sqrt{1-x^2}) \]
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