দেওয়া আছে, $$ \begin{aligned} \sin \alpha - \sin \beta &= 2 \quad \dots (i) \\ \cos \alpha + \cos \beta &= 3 \quad \dots (ii) \end{aligned} $$ (i) নং কে (ii) নং দ্বারা ভাগ করে পাই, $$ \begin{aligned} \frac{\sin \alpha - \sin \beta}{\cos \alpha + \cos \beta} &= \frac{2}{3} \\ \Rightarrow \frac{2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}}{2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}} &= \frac{2}{3} \\ \Rightarrow \tan \frac{\alpha-\beta}{2} &= \frac{2}{3} \end{aligned} $$ আমরা জানি, $$ \begin{aligned} \cos(\alpha-\beta) &= \frac{1 - \tan^2 \frac{\alpha-\beta}{2}}{1 + \tan^2 \frac{\alpha-\beta}{2}} \\ &= \frac{1 - (\frac{2}{3})^2}{1 + (\frac{2}{3})^2} \\ &= \frac{1 - \frac{4}{9}}{1 + \frac{4}{9}} \\ &= \frac{\frac{5}{9}}{\frac{13}{9}} \\ &= \frac{5}{13} \end{aligned} $$