If \(\sin \alpha+\sin \beta=\frac{\sqrt{2}}{\sqrt{3}}\) and \(\cos \alpha+\cos \beta=\frac{1}{\sqrt{3}}\), then the value of \(\left(20 \cos \left(\frac{\alpha-\beta}{2}\right)\right)^{2}\) is

The required square is $100$. The sum of the two given squares is $\frac23+\frac13=1$, and the sum-to-product identities make this $4\cos^2\bigl(\frac{\alpha-\beta}{2}\bigr)$. Multiplying $\frac14$ by $20^2$ gives $100$, while dividing $1$ by just $2$ gives $200$ after scaling.