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Solution

Use identities $\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B$ $\cos(A+B)\cos(A-B) = \cos^2 A - \cos^2 B$ $\displaystyle \frac{\cos 60^\circ \cos 36^\circ}{\sin 30^\circ \sin 18^\circ} = \frac{\frac{\sqrt{5}+1}{4}}{\frac{\sqrt{5}-1}{4}} = \frac{\sqrt{5}+1}{\sqrt{5}-1}$ $= \frac{(\sqrt{5}+1)^2}{5-1} = \frac{6+2\sqrt{5}}{4} = \frac{3+\sqrt{5}}{2}$ $\Rightarrow \alpha = 3,; \beta = 1$ $\Rightarrow \alpha + \beta = 4$