Explanation

(c)
Given that, $\sin^{- 1}{\frac{1}{3} + \sin^{- 1}{\frac{2}{3} = \sin^{1}x}}$
$\Rightarrow \ \sin^{- 1}{\left( \frac{1}{3}\sqrt{1 - \frac{4}{9}} + \frac{2}{3}\sqrt{1 - \frac{1}{9}} \right) = \sin^{- 1}x}$
$\Rightarrow \sin^{- 1}\left( \frac{1}{3}.\frac{\sqrt{5}}{3} + \frac{2}{3}.\frac{\sqrt{8}}{3} \right) = \sin^{- 1}x$
$\Rightarrow \ \ \sin^{- 1}{\left( \frac{\sqrt{5} + 4\sqrt{2}}{9} \right) = \sin^{- 1}x}$
$\therefore\ \ \ \ x = \left( \frac{\sqrt{5} + 4\sqrt{2}}{9} \right)$