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

$\begin{array}{rcl}\sin \mathrm{A}+\sin 2\mathrm{A}& =& \mathrm{x}, \cos \mathrm{A}+\cos 2\mathrm{A}=\mathrm{y}\\ {\mathrm{x}}^2+{\mathrm{y}}^2& =& {\sin }^2\mathrm{a}+{\sin }^{2}2\mathrm{A}+2\sin \mathrm{A} \sin 2\mathrm{A}\\ & & +{\cos }^2\mathrm{A}+{\cos }^{2}2\mathrm{A}+2\cos \mathrm{A} \cos 2\mathrm{A}\\ & =& 1+1+2(\cos 2\mathrm{A} \cos \mathrm{A}+\sin 2\mathrm{A} \sin \mathrm{A})\\ & =& 2+2\cos (2\mathrm{A}-\mathrm{A})\\ {\mathrm{x}}^2+{\mathrm{y}}^2-2& =& 2\cos \mathrm{A}\\ \cos \mathrm{A}& =& \frac{1}{2}\left({\mathrm{x}}^2+{\mathrm{y}}^2-\mathrm{z}\right)\end{array}$