$\operatorname{Cos}(\alpha \pm \beta )=\operatorname{Cos}\alpha \operatorname{Cos}\beta \mp \operatorname{Sin}\alpha \operatorname{Sin}\beta $ 

با استفاده از اتحادهای مثلثاتی بالا داریم:

$\operatorname{Cos}(x+\frac{\pi }{3})\operatorname{Cos}(x-\frac{\pi }{3})=(\operatorname{Cos}x\operatorname{Cos}\frac{\pi }{3}-\operatorname{Sin}x\operatorname{Sin}\frac{\pi }{3})(\operatorname{Cos}x\operatorname{Cos}\frac{\pi }{3}+\operatorname{Sin}x\operatorname{Sin}\frac{\pi }{3})$

$=\frac{1}{4}{{\operatorname{Cos}}^{2}}x-\frac{3}{4}{{\operatorname{Sin}}^{2}}x=-\frac{1}{2}\xrightarrow{\times 4}{{\operatorname{Cos}}^{2}}x-3{{\operatorname{Sin}}^{2}}x=-2\xrightarrow{{{\operatorname{Sin}}^{2}}x=1-{{\operatorname{Cos}}^{2}}x}{{\operatorname{Cos}}^{2}}x-3(1-{{\operatorname{Cos}}^{2}}x)=-2$

$\Rightarrow {{\operatorname{Cos}}^{2}}x=\frac{1}{4}={{\operatorname{Cos}}^{2}}\frac{\pi }{3}\Rightarrow x=k\pi \pm \frac{\pi }{3}$