দেওয়া আছে, $$ \begin{aligned} \sin \alpha + \sin \beta &= a \quad \dots(1) \\ \cos \alpha + \cos \beta &= b \quad \dots(2) \end{aligned} $$ (1) ও (2) বর্গ করে যোগ করলে, $$ \begin{aligned} (\sin \alpha + \sin \beta)^2 + (\cos \alpha + \cos \beta)^2 &= a^2 + b^2 \\ \Rightarrow (\sin^2 \alpha + \cos^2 \alpha) + (\sin^2 \beta + \cos^2 \beta) + 2(\cos \alpha \cos \beta + \sin \alpha \sin \beta) &= a^2 + b^2 \\ \Rightarrow 1 + 1 + 2 \cos(\alpha - \beta) &= a^2 + b^2 \\ \Rightarrow 2 \cos(\alpha - \beta) &= a^2 + b^2 - 2 \\ \Rightarrow \cos(\alpha - \beta) &= \frac{a^2 + b^2 - 2}{2} \end{aligned} $$