If $ an ^{-1} \sqrt{x^2+x}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2}$, then the value of $x$ is

Question

If $	an ^{-1} \sqrt{x^2+x}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2}$, then the value of $x$ is

Aicharya · Accepted Answer

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

If $\tan ^{-1} \sqrt{x^2+x}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2}$, then the value of $x$ is

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