Given that,

$A\left(n\right)={\sin }^4\alpha +{\cos }^4\alpha$

And

$A\left(1\right)A\left(4\right)+A\left(2\right)A\left(5\right)=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha +{\cos }^4\alpha \right)+\left({\sin }^2\alpha +{\cos }^2\alpha \right)\left({\sin }^5\alpha +{\cos }^5\alpha \right)$

$\Rightarrow A\left(1\right)A\left(4\right)+A\left(2\right)A\left(5\right)=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha +{\cos }^4\alpha \right)+\left({\sin }^5\alpha +{\cos }^5\alpha \right)$

Consider,

$\Rightarrow A\left(1\right)A\left(6\right)+A\left(2\right)A\left(3\right)=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^6\alpha +{\cos }^6\alpha \right)+\left({\sin }^2\alpha +{\cos }^2\alpha \right)\left({\sin }^3\alpha +{\cos }^3\alpha \right)$

$=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha {\sin }^2\alpha +{\cos }^4\alpha {\cos }^2\alpha \right)+\left({\sin }^2\alpha +{\cos }^2\alpha \right)\left({\sin }^3\alpha +{\cos }^3\alpha \right)$

$=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha \left(1-{\cos }^2\alpha \right)+{\cos }^4\alpha \left(1-{\sin }^2\alpha \right)\right)+\left({\sin }^3\alpha +{\cos }^3\alpha \right)$

$=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha -{\sin }^4\alpha {\cos }^2\alpha +{\cos }^4\alpha -{\cos }^4\alpha {\sin }^2\alpha \right)+\left({\sin }^3\alpha +{\cos }^3\alpha \right)$

$=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha +{\cos }^4\alpha \right)-\left(\sin \alpha +\cos \alpha \right){\sin }^4\alpha {\cos }^2\alpha -\left(\sin \alpha +\cos \alpha \right){\cos }^4\alpha {\sin }^2\alpha +\left({\sin }^3\alpha +{\cos }^3\alpha \right)$

$=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha +{\cos }^4\alpha \right)-\left(\sin \alpha +\cos \alpha \right){\sin }^2\alpha {\cos }^2\alpha \left({\sin }^2\alpha +{\cos }^2\alpha \right)+\left({\sin }^3\alpha +{\cos }^3\alpha \right)$

$=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha +{\cos }^4\alpha \right)-\left(\sin \alpha +\cos \alpha \right)\left({\sin }^2\alpha \right)\left({\cos }^2\alpha \right)+\left({\sin }^3\alpha +{\cos }^3\alpha \right)$

$=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha +{\cos }^4\alpha \right)-{\sin }^3\alpha {\cos }^2\alpha -{\cos }^3\alpha {\sin }^2\alpha +\left({\sin }^3\alpha +{\cos }^3\alpha \right)$

$=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha +{\cos }^4\alpha \right)+{\sin }^3\alpha \left(1-{\cos }^2\alpha \right)+{\cos }^3\alpha \left(1-{\sin }^2\alpha \right)$

$=\left(\sin \alpha +\cos \alpha \right)\left({\sin }^4\alpha +{\cos }^4\alpha \right)+\left({\sin }^5\alpha +{\cos }^5\alpha \right)$

$=A\left(1\right)A\left(4\right)+A\left(2\right)A\left(5\right)$

$\therefore A\left(1\right)A\left(4\right)+A\left(2\right)A\left(5\right)=A\left(1\right)A\left(6\right)+A\left(2\right)A\left(3\right)$