$_{\operatorname{Sin}(\pi \pm \alpha )=\mp \operatorname{Sin}\alpha ,\operatorname{Cos}(\pi \pm \alpha )=-\operatorname{Cos}\alpha }^{\operatorname{Sin}(\frac{\pi }{2}\pm \alpha )=\operatorname{Cos}\alpha ,\operatorname{Cos}(\frac{\pi }{2}\pm \alpha )=\mp \operatorname{Sin}\alpha }$  طبق فرض $\alpha $ و $\beta $ مکمل‌اند، پس: $\alpha +\beta =\pi \Rightarrow \beta =\pi -\alpha $  اكنون با استفاده از نكات بالا داريم: $\operatorname{Sin}(\frac{\pi }{2}+\beta )=\operatorname{Cos}\beta =\operatorname{Cos}(\pi -\alpha )=-\operatorname{Cos}\alpha $