If [ ^ -1 ( x + 1)] = ( ^ -1 x), then find x. - Vidyadip

Question

$\text{If} \sin [\cot^{-1} ( x + 1)] = \cos(\tan^{-1}x), \text{then find x}.$

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Answer

$\text{Writing} \cot^{-1} (\text{x + 1}) = \sin^{-1} \frac{1}{\sqrt{1 + ( \text{x + 1})^{2}}}$
$\text{and} \tan^{-1}\text{x} = \cos^{-1} \frac{1}{\sqrt{1 + \text{x}^{2}}}$
$\therefore \sin \bigg(\sin^{-1} \frac{1}{\sqrt{1+{\text{(x + 1)}}^{2}}}\bigg) = \cos \bigg(\cos^{-1} \frac{1}{\sqrt{1 + \text{x}^{2}}}\bigg)$
$1 + \text{x}^{2} + 2\text{x} + 1 = 1 + \text{x}^{2} \Rightarrow \text{x} = -\frac{1}{2}$

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