Question:
The number of real roots of the equation $\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{4}$ is :
Correct Option: , 4
Solution:
$\tan ^{-1} \sqrt{x^{2}+x}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{4}$
For equation to be defined,
$x^{2}+x \geq 0$
$\Rightarrow \quad x^{2}+x+1 \geq 1$
$\therefore \quad$ only possibility that the equation is defined
$x^{2}+x=0 \quad \Rightarrow x=0 ; x=-1$
None of these values satisfy
$\therefore$ No of roots $=0$
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