The number of real roots of the equation ^ - 1 x(x + 1) + ^ - 1 x ^...

We have,

$\tan^{- 1}\sqrt{{x}^{2} + x} + \sin^{- 1}\sqrt{{x}^{2} + x + 1} = \dfrac{\pi}{4}$

For equation to be defined,

for $\tan^{- 1}\sqrt{x\left(x + 1\right)}$,

${x}^{2} + x \ge 0$

$⇒ {x}^{2} + x + 1 \ge 1 . . . . \left(1\right)$

And, from domain of $\sin^{- 1}\sqrt{{x}^{2} + x + 1}$, ${x}^{2} + x + 1 \le 1 . . . . \left(2\right)</p><p>$∴ from \left(1\right) \text{and } \left(2\right) only possibility that the equation is defined is

${x}^{2} + x = 0$

$⇒ x = 0, - 1$

at $x = 0$, $\tan^{- 1}\sqrt{x\left(x + 1\right)} +$\sin^{- 1}\sqrt{{x}^{2}+ x + 1} = 0 + \dfrac{\pi}{2} \ne \dfrac{\pi}{4}

and at $x = - 1$, $\tan^{- 1}\sqrt{x\left(x + 1\right)} +$\sin^{- 1}\sqrt{{x}^{2}+ x + 1} = 0 + \dfrac{\pi}{2} \ne \dfrac{\pi}{4}

None$of these values satisfy</p><p>∴$ Number of roots$= 0$