$\text{ The absolute value of }x\text{ for which }\sin \left({\cot }^{-1}(1+x)\right)=\cos \left({\tan }^{-1}x\right) is$

$\begin{array}{l}\sin \left({\cot }^{-1}(x+1)\right)=\cos \left({\tan }^{-1}x\right)\\ \Rightarrow \cos \left(\pi /2-{\cot }^{-1}(x+1)\right)=\cos \left({\tan }^{-1}x\right)\\ \\ \Rightarrow \pi /2-{\cot }^{-1}(\text{x}+1)=\pm {\tan }^{-1}\text{x}={\tan }^{-1}(\pm \text{x})\\ \Rightarrow \pi /2={\tan }^{-1}(\pm x)+{\cot }^{-1}(x+1)\\ \Rightarrow x+1=\pm x\Rightarrow 2x+1=0\\ \text{⇒x}=-\frac{1}{2}\end{array}$