Using the expansions of \(\sin ( 3 x + 2 x )\) and \(\sin ( 3 x - 2 x )\), show that
$$\frac { 1 } { 2 } ( \sin 5 x + \sin x ) \equiv \sin 3 x \cos 2 x$$
Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin 3 x \cos 2 x \mathrm {~d} x = \frac { 1 } { 5 } ( 3 - \sqrt { 2 } )\).