If $\mathrm{a}=\mathrm{Cos\theta }-\mathrm{Sin\theta }; \mathrm{b}={\mathrm{Cos}}^3\mathrm{\theta }-{\mathrm{Sin}}^3\mathrm{\theta };$ then 

$\mathrm{a}=\mathrm{Cos\theta }-\mathrm{Sin\theta }; \mathrm{b}={\mathrm{Cos}}^3\mathrm{\theta }-{\mathrm{Sin}}^3\mathrm{\theta };$

$\begin{array}{rcl}{\mathrm{a}}^2& =& 1-2 \mathrm{sin\theta } \mathrm{cos\theta }\\ \mathrm{sin\theta } \mathrm{cos\theta }& =& \frac{1-{\mathrm{a}}^2}{2}\\ \mathrm{b}& =& {\cos }^3\mathrm{\theta }-{\sin }^3\mathrm{\theta }\\ & =& (\mathrm{cos\theta }-\mathrm{sin\theta })(1+\mathrm{sin\theta } \mathrm{cos\theta })\\ \mathrm{b}& =& \mathrm{a}\left(1+\frac{1-{\mathrm{a}}^2}{2}\right)\\ 2\mathrm{b}& =& \mathrm{a}(3-{\mathrm{a}}^2)\\ {\mathrm{a}}^3-3\mathrm{a}+2\mathrm{b}& =& 0\end{array}$