نکته: $\operatorname{Cos}(-x)=\operatorname{Cos}x,\operatorname{Cos}(\alpha \pm \beta )=\operatorname{Cos}\alpha \operatorname{Cos}\beta \mp \operatorname{Sin}\alpha \operatorname{Sin}\beta $
با استفاده از نكته میتوان عبارت $A$ را بهصورت زير ساده كرد:
$A=\operatorname{Cos}\overbrace{(\alpha -\beta )}^{x}\operatorname{Cos}\overbrace{(\alpha +\beta )}^{y}+\operatorname{Sin}\overbrace{(\alpha -\beta )}^{x}\operatorname{Sin}\overbrace{(\alpha +\beta )}^{y}=\operatorname{Cos}((\alpha -\beta )-(\alpha +\beta ))=\operatorname{Cos}(-2\beta )=\operatorname{Cos}2\beta $