$\begin{aligned} & \tan ^{-1} \sqrt{x^2+x}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2} \\ & \Rightarrow \tan ^{-1} \sqrt{x^2+x}=\frac{\pi}{2}-\sin ^{-1} \sqrt{x^2+x+1}=\cos ^{-1} \sqrt{x^2+x+1} \\ & \Rightarrow \tan ^{-1} \sqrt{x^2+x}=\tan ^{-1} \frac{\sqrt{-\left(x^2+x\right)}}{\sqrt{x^2+x+1}} ; \sqrt{-\left(x^2+x\right)} \\ & \Rightarrow x^2+x=\frac{-\left(x^2+x\right)}{x^2+x+1} \\ & \Rightarrow\left(x^2+x\right)\left(x^2+x+1\right)=-\left(x^2+x\right)\left[\text { let } x^2+x=y\right] \\ & \Rightarrow y(y+1)=-y \\ & \Rightarrow y^2+2 y=0 \\ & \Rightarrow y(y+2)=0 \\ & \Rightarrow y=0 \text { or } y=-2 \\ & \Rightarrow x^2+x=0 \text { or } x^2+x=-2 \\ & \Rightarrow x(x+1)=0 \text { or } x^2+x+2=0 \\ & \Rightarrow(x=0,-1) \text { or } \text { (No solution) }\end{aligned}$