• # question_answer If $a\,{{\cos }^{3}}\alpha +3a\,\cos \alpha \,{{\sin }^{2}}\alpha =m$ and$a\,{{\sin }^{3}}\alpha +3a\,{{\cos }^{2}}\alpha \sin \alpha =n,$ then ${{(m+n)}^{2/3}}+{{(m-n)}^{2/3}}$ is equal to A) $2{{a}^{2}}$ B) $2{{a}^{1/3}}$ C) $2{{a}^{2/3}}$ D) $2{{a}^{3}}$

Adding and subtracting the given relation, we get $(m+n)=a{{\cos }^{3}}\alpha +3a\cos \alpha \,{{\sin }^{2}}\alpha$$+3a{{\cos }^{2}}\alpha .\sin \alpha +a{{\sin }^{3}}\alpha$ $=a{{(\cos \alpha +\sin \alpha )}^{3}}$ and similarly $(m-n)=a\,\,{{(\cos \alpha -\sin \alpha )}^{3}}$ Thus, ${{(m+n)}^{2/3}}+{{(m-n)}^{2/3}}$ $={{a}^{2/3}}{{\{\cos \alpha +\sin \alpha )}^{2}}+{{(\cos \alpha -\sin \alpha )}^{2}}\}$ $={{a}^{2/3}}\{2({{\cos }^{2}}\alpha +{{\sin }^{2}}\alpha )\}=2{{a}^{2/3}}$.