If sin left(cot ^{-1}(x+1)right)=cos left(tan | vedclass

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(A) Given equation: $\sin \left(\cot ^{-1}(x+1)ight)=\cos \left(	an ^{-1} xight)$ Let $\cot ^{-1}(x+1) = 	heta$,then $\cot 	heta = x+1$. Using

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$-\frac{1}{2}$

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(A) Given equation: $\sin \left(\cot ^{-1}(x+1)ight)=\cos \left(	an ^{-1} xight)$ Let $\cot ^{-1}(x+1) = 	heta$,then $\cot 	heta = x+1$. Using the identity $\sin 	heta = \frac{1}{\sqrt{1+\cot^2 	heta}}$,we get $\sin 	heta = \frac{1}{\sqrt{1+(x+1)^2}} = \frac{1}{\sqrt{x^2+2x+2}}$. Let $	an ^{-1} x = \phi$,then $	an \phi = x$. Using the identity $\cos \phi = \frac{1}{\sqrt{1+	an^2 \phi}}$,we get $\cos \phi = \frac{1}{\sqrt{1+x^2}}$. Equating both sides: $\frac{1}{\sqrt{x^2+2x+2}} = \frac{1}{\sqrt{x^2+1}}$. Squaring both sides: $x^2+1 = x^2+2x+2$. $2x = -1 \Rightarrow x = -\frac{1}{2}$.

If $\sin \left(\cot ^{-1}(x+1)\right)=\cos \left(\tan ^{-1} x\right)$,then the value of $x$ is equal to

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