$\begin{array}{l} \left( {\sin x + \cos x} \right)\left( {1 - \dfrac{1}{2}\sin 2x} \right)\\ = \left( {\sin x + \cos x} \right)\left( {{{\sin }^2}x + {{\cos }^2}x - \dfrac{1}{2}\left( {2\sin x.\cos x} \right)} \right)\\ = \left( {\sin x + \cos x} \right)\left( {{{\sin }^2}x + {{\cos }^2}x - \sin x\cos x} \right)\\ = {\sin ^3}x + {\cos ^3}x \end{array}$
