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Prove that sin^(-1) (2xsqrt(1-x^2))=2cos...

Prove that `sin^(-1) (2xsqrt(1-x^2))=2cos^(-1)x,1/sqrt2 le x le 1`

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To prove that \( \sin^{-1}(2x\sqrt{1-x^2}) = 2\cos^{-1}(x) \) for \( \frac{1}{\sqrt{2}} \leq x \leq 1 \), we will follow these steps: ### Step 1: Substitute \( x \) with \( \sin \theta \) Let \( x = \sin \theta \). Then, we have: \[ \sqrt{1 - x^2} = \sqrt{1 - \sin^2 \theta} = \cos \theta \] This gives us: ...
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