$\begin{array}{l} {\sin ^3}\dfrac{x}{3} = \dfrac{1}{4}\left( {3\sin \dfrac{x}{3} - \sin x} \right)\\ 3{\sin ^3}\dfrac{x}{9} = \dfrac{3}{4}\left( {3\sin \dfrac{x}{9} - \sin \dfrac{x}{3}} \right)\\ 9{\sin ^3}\dfrac{x}{{27}} = \dfrac{9}{4}\left( {3\sin \dfrac{x}{{27}} - \sin \dfrac{x}{9}} \right)\\ ....\\ {3^{2019}}{\sin ^3}\dfrac{x}{{{3^{2020}}}} = \dfrac{{{3^{2019}}}}{4}\left( {3\sin \dfrac{x}{{{3^{2020}}}} - \sin \dfrac{x}{{{3^{2019}}}}} \right)\\ {3^{2020}}{\sin ^3}\dfrac{x}{{{3^{2021}}}} = \dfrac{{{3^{2020}}}}{4}\left( {3\sin \dfrac{x}{{{3^{2021}}}} - \sin \dfrac{x}{{{3^{2020}}}}} \right)\\ \Rightarrow A = - \dfrac{1}{4}\sin x + \dfrac{{{3^{2021}}}}{4}\sin \dfrac{x}{{{3^{2021}}}} \end{array}$
