If $\sin \left({\cot }^{-1}\left(1-x\right)\right)=\cos \left({\tan }^{-1}\left(-x\right)\right)$, then $x$ is

• A

  $1$
• Correct answer
  B

  $\frac{1}{2}$
• C

  $0$
• D

  $-\frac{1}{2}$

Solution

$\sin \left({\cot }^{-1}\left(1-x\right)\right)=\cos \left({\tan }^{-1}\left(-x\right)\right)$
Let, ${\cot }^{-1}\left(1-x\right)=\theta$
$\Rightarrow \cot \theta =\left(1-x\right)$

$\Rightarrow \sin \theta =\frac{1}{\sqrt{2-2x+x^2}}$
Let, ${\tan }^{-1}\left(-x\right)=\varphi$
$\Rightarrow \tan \varphi =-x\Rightarrow \cos \varphi =\frac{1}{\sqrt{1+x^2}}$

$\therefore \sin \theta =\cos \varphi$
$\Rightarrow \frac{1}{\sqrt{2-2x+x^2}}=\frac{1}{\sqrt{1+x^2}}$
$\Rightarrow 1+x^2=2-2x+x^2$
$\Rightarrow x=\frac{1}{2}$

Hence, the correct answer is option: 'B'.