\({\sin ^3}x + {\cos ^3}x = \sin x-\cos x\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{\sin ^3}x + {\cos ^3}x = \sin x \left( {{{\sin }^2}x + {{\cos }^2}x} \right)-\cos x\left( {{{\sin }^2}x + {{\cos }^2}x} \right)\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{\sin ^3}x + {\cos ^3}x = {\sin ^3}x + \sin x{\cos ^2}x-\cos x{\sin ^2}x-{\cos ^3}x\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\cos x{\sin ^2}x-\sin x{\cos ^2}x + 2{\cos ^3}x = 0\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\cos x\left( {{{\sin }^2}x-\sin x\cos x + 2{{\cos }^2}x} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos x = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{{\sin }^2}x-\sin x\cos x + 2{{\cos }^2}x = 0.}\end{array}} \right.\)
\(\cos x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \dfrac{\pi }{2} + \pi k,\,\,\,\,\,\,\,\,k\, \in \,Z.\)
Уравнение \({\sin ^2}x-\sin x\cos x + 2{\cos ^2}x = 0\) является однородным тригонометрическим уравнением второй степени:
\({\sin ^2}x-\sin x\cos x + 2{\cos ^2}x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{t}}{{\rm{g}}^2}x-{\rm{tg}}\,x + 2 = 0.\)
Пусть \({\rm{tg}}\,x = t,\,\,\,\,\,\,\,t\, \in \,R\). Тогда:
\({t^2}-t + 2 = 0;\,\,\,\,\,\,\,\,\,\,\,\,D = 1-8 = -7 < 0.\)
Следовательно, последнее уравнение не имеет решений.
Ответ: \(\dfrac{\pi }{2} + \pi k;\;\,\,\,\;k \in Z.\)