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

با توجه به نکته، $\operatorname{Sin}{{75}^{{}^\circ }}$ و $\operatorname{Sin}{{15}^{{}^\circ }}$ را به‌ صورت‌های زير نوشته و عبارت را ساده می‌كنيم:

$\operatorname{Sin}{{75}^{{}^\circ }}+\operatorname{Sin}{{15}^{{}^\circ }}=\operatorname{Sin}({{45}^{{}^\circ }}+{{30}^{{}^\circ }})+\operatorname{Sin}({{45}^{{}^\circ }}-{{30}^{{}^\circ }})=\operatorname{Sin}{{45}^{{}^\circ }}\operatorname{Cos}{{30}^{{}^\circ }}+\operatorname{Sin}{{30}^{{}^\circ }}\operatorname{Cos}{{45}^{{}^\circ }}+\operatorname{Sin}{{45}^{{}^\circ }}\operatorname{Cos}{{30}^{{}^\circ }}-\operatorname{Sin}{{30}^{{}^\circ }}\operatorname{Cos}{{45}^{{}^\circ }}=2\operatorname{Sin}{{45}^{{}^\circ }}\operatorname{Cos}{{30}^{{}^\circ }}=2\times \frac{\sqrt{2}}{2}\times \frac{\sqrt{3}}{2}=\frac{\sqrt{6}}{2}$