Explanation

(c)
$\sin{\left\lbrack 2\cos^{- 1}\frac{\sqrt{5}}{3} \right\rbrack = \sin\left\lbrack \cos^{- 1}\left\{ 2.\left( \frac{\sqrt{5}}{3} \right)^{2} - 1 \right\} \right\rbrack}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lbrack\because 2\cos^{- 1}{x = \cos^{- 1}{(2x^{2} - 1)}}\rbrack$
$= \sin\left\lbrack \cos^{- 1}\left( \frac{1}{9} \right) \right\rbrack$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sin\left\lbrack \sin^{- 1}\sqrt{1 - \left( \frac{1}{9} \right)^{2}} \right\rbrack$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lbrack\because\cos^{- 1}{x = \sin^{- 1}{(\sqrt{1 - x^{2}})}}\rbrack$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{4\sqrt{5}}{9}$