Воспользуемся основным тригонометрическим тождеством:
\({\sin ^2}\alpha + {\cos ^2}\alpha = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\sin ^2}\alpha + {\left( {\dfrac{7}{{25}}} \right)^2} = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\sin ^2}\alpha = \dfrac{{576}}{{625}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\sin \alpha = \dfrac{{24}}{{25}},\,\,\,}\\{\sin \alpha = -\dfrac{{24}}{{25}}.}\end{array}} \right.\)
Так как \(\dfrac{{3\pi }}{2} < \alpha < 2\pi \) (IV четверть), то \(\sin \alpha < 0\), то есть \(\sin \alpha = -\dfrac{{24}}{{25}}.\)
\({\sin ^2}\beta + {\cos ^2}\beta = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\left( {\dfrac{{24}}{{25}}} \right)^2} + {\cos ^2}\beta = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\cos ^2}\beta = \dfrac{{49}}{{625}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos \beta = \dfrac{7}{{25}},\,\,\,}\\{\cos \beta = -\dfrac{7}{{25}}.}\end{array}} \right.\)
Так как \(-2\pi < \beta < -\dfrac{{3\pi }}{2}\) (I четверть), то \(\cos \beta > 0\), то есть \(\cos \beta = \dfrac{7}{{25}}.\)
\(25\sin \left( {\alpha -\beta } \right) = 25\left( {\sin \alpha \cos \beta -\cos \alpha \sin \beta } \right) = 25\left( {-\dfrac{{24}}{{25}} \cdot \dfrac{7}{{25}}-\dfrac{7}{{25}} \cdot \dfrac{{24}}{{25}}} \right) = -\dfrac{{336}}{{25}} = -13,44.\)
Ответ: \(-13,44.\)