\( tan^{-1}\sqrt{x^2+x}\:exist\:only\:if \:x^2+x\geq\:0\)
and \( \: sin^{-1} \sqrt{x^2+x+1} \) exist only if \( -1 \leq\:\sqrt{x^2+x+1} \leq 1\)
\(\Rightarrow\) \( x^2+x \geq 0 \: and \: 0 \leq x^2+x+1\leq 1\)
\(\Rightarrow\:x^2+x\geq\:0\:and\:x^2+x+1\leq\:1 \)
This is possible only if \(x^2+x=0\)
solving which we get \( x=0, -1 \)
Both the values satisfy the given equation.