If $\sin A+\sin 2A=x,\cos A+\cos 2A=y$ then $\cos A=$

$\Rightarrow x^2={\left[\sin A+\sin 2A\right]}^2$

$\Rightarrow x^2={\sin }^{2}A+{\sin }^{2}2A+2\sin A.\sin 2A$

$y=\cos A+\cos 2A$

$\Rightarrow y^2={\cos }^{2}A+{\cos }^{2}2A+2\cos A.\cos 2A$

$x^2+y^2=\left[{\sin }^{2}A+{\cos }^{2}A\right]+\left[{\cos }^{2}2A+{\sin }^{2}2A\right]+2\sin A.\sin 2A+2\cos A.\cos 2A$

$=2+2\sin A.2\sin A.\cos A+2\cos A\left[2{\cos }^{2}A-1\right]$

$=2+4{\sin }^{2}A.\cos A+4{\cos }^{3}A-2\cos A$

$=2+4\cos A\left[{\sin }^{2}A+{\cos }^{2}A\right]-2\cos A$

$=2+4\cos A-2\cos A$

$\Rightarrow x^2+y^2=2+2\cos A$

$\Rightarrow x^2+y^2-2=2\cos A$

$\Rightarrow \frac{1}{2}\left[x^2+y^2-2\right]=\cos A$