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: Let \( x = \sin(\theta) \)**
We start by substituting \( x \) with \( \sin(\theta) \). Thus, we have:
\[
\sin^{-1}(2x\sqrt{1-x^2}) = \sin^{-1}(2\sin(\theta)\sqrt{1-\sin^2(\theta)})
...