The value of x for which sin ( cot - 1 ( 1 + x ) ) = cos ( tan - 1 x ) is [2004]

Question

The value of $x$ for which $\sin (\cot^{- 1} (1 + x)) = \cos (\tan^{- 1} x)$ is [2004]

Smart Achievers · Accepted Answer

(4)

$\sin [\cot^{- 1} (1 + x)] = \cos (\tan^{- 1} x)$

$\Rightarrow \sin [\sin^{- 1} (\frac{1}{\sqrt{1 + {(1 + x)}^{2}}})] = \cos [\cos^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})]$

$\Rightarrow \frac{1}{\sqrt{1 + {(1 + x)}^{2}}} = \frac{1}{\sqrt{1 + x^{2}}}$

$\Rightarrow 1 + 1 + 2 x + x^{2} = 1 + x^{2}$

$\Rightarrow 2 x + 1 = 0$

$\Rightarrow x = - \frac{1}{2}$

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