MCQ
${{\sin }^{-1}}\left[ x\sqrt{1-x}-\sqrt{x}\sqrt{1-{{x}^{2}}} \right]=......$
- A${{\sin }^{-1}}\sqrt{x}+{{\sin }^{-1}}x$
- ✓${{\sin }^{-1}}x-{{\sin }^{-1}}\sqrt{x}$
- C${{\sin }^{-1}}\sqrt{x}-{{\sin }^{-1}}x$
- Dઆમાંથી એક પણ નહિં.
$\sin^{-1} \left[ x\sqrt {1-x} - \sqrt{x} \sqrt{1-x^2}\right]$
અહી $x = \sin \theta$ અને $\sqrt{x} = \sin \alpha$
$\theta = \sin^{-1}x$ અને $\alpha = \sin^{-1} \sqrt {x}$
$= \sin^{-1} (\sin \theta \sqrt {1-\sin^2 \alpha} - \sin \alpha \sqrt {1-\sin^2 \theta}$
$= \sin^{-1} (\sin \theta\cos \alpha - \sin \alpha \cos \theta) $
$= \sin^{-1} (\sin (\theta - \alpha ))$
$= \theta - \alpha$
$=\sin^{-1} x - \sin^{-1} \sqrt{x} (0 \leq \theta, \alpha\leq \frac{\pi}{2})$
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