Prove that (\cos A+\sin A)(\cos 2A+\sin 2A)=\cos A+\sin 3A

Prove that $\left(\mathrm{cos}A+\mathrm{sin}A\right)\left(\mathrm{cos}2A+\mathrm{sin}2A\right)=\mathrm{cos}A+\mathrm{sin}3A$
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Use following facts:
$\mathrm{cos}A\mathrm{sin}2A+\mathrm{sin}A\mathrm{cos}2A=\mathrm{sin}\left(A+2A\right)=\mathrm{sin}3A$
$\mathrm{cos}A\mathrm{cos}2A+\mathrm{sin}A\mathrm{sin}2A=\mathrm{cos}\left(2A-A\right)=\mathrm{cos}A$