Explanation

(c)
Given, $\tan^{- 1}{\sqrt{x(x + 1)} = \frac{\pi}{2} - \sin^{- 1}\sqrt{x^{2} + x + 1}}$
$\Rightarrow \ \cos^{- 1}{\frac{1}{\sqrt{\left( x^{2} + x \right)^{2} + 1}} = \cos^{- 1}\sqrt{x^{2} + x + 1}}$
$\Rightarrow \ \frac{1}{\sqrt{\left( x^{2} + x \right)^{2} + 1}} = \sqrt{x^{2} + x + 1}$
$\Rightarrow \ \ \ 1 = \left( x^{2} + x + 1 \right)\lbrack\left( x^{2} + x \right)^{2} + 1\rbrack$
$\Rightarrow \ \ \ \left( x^{2} + x \right)^{3} + \left( x^{2} + x \right)^{2} + \left( x^{2} + x \right) + 1 = 1$
$\Rightarrow \ \ \left( x^{2} + x \right)\left\{ \left( x^{2} + x \right)^{2} + \left( x^{2} + x \right) + 1 \right\} = 0$
$\Rightarrow \ \ x^{2} + x = 0$
$\Rightarrow \ \ \ x = 0,\ - 1$