7. The minimum value of the algebraic expression $9 \cos ^{2} \alpha-10 \cos \alpha \sin \beta-8 \cos \beta \sin \alpha+17$ is $\qquad$
7. 1 Solution: Transform the original expression into $25 \cos ^{2} \alpha+16 \sin ^{2} \alpha-10 \cos \alpha \sin \beta-8 \sin \alpha \cos \beta+\cos ^{2} \beta+\sin ^{2} \beta=(5 \cos \alpha- \sin \beta)^{2}+(4 \sin \alpha-\cos \beta)^{2}$.

This represents the square of the minimum distance between a point on the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and a point on the circle $x^{2}+y^{2}=1$. By geometric interpretation, the minimum squared distance is 1.