দেওয়া আছে, $$ \begin{aligned} \cos \alpha + \sin \beta &= 0 \quad \dots(1) \\ \sin \alpha - \cos \beta &= 1 \quad \dots(2) \end{aligned} $$ সমীকরণ (1) ও (2) বর্গ করে যোগ করলে পাই, $$ \begin{aligned} (\cos \alpha + \sin \beta)^2 + (\sin \alpha - \cos \beta)^2 &= 0^2 + 1^2 \\ \Rightarrow \cos^2 \alpha + 2\cos \alpha \sin \beta + \sin^2 \beta + \sin^2 \alpha - 2\sin \alpha \cos \beta + \cos^2 \beta &= 1 \\ \Rightarrow (\sin^2 \alpha + \cos^2 \alpha) + (\sin^2 \beta + \cos^2 \beta) - 2(\sin \alpha \cos \beta - \cos \alpha \sin \beta) &= 1 \\ \Rightarrow 1 + 1 - 2\sin(\alpha - \beta) &= 1 \\ \Rightarrow 2\sin(\alpha - \beta) &= 1 \\ \Rightarrow \sin(\alpha - \beta) &= \frac{1}{2} \\ \Rightarrow \alpha - \beta &= 30^{\circ} \end{aligned} $$