Gauss-Bonnet formula and the geometry convexity of cos(x~(1/2))

By making use of Gauss-Bonnet Formula,the author in this paper not only explains the relationship between the sides and angles of geodesic triangles in several constant curvature spaces,but also proves it with the geometry convexity of cosx～(1/2).Compared with the Pythagorean theorem in Euclid space,It manifests completeness in both form and content.At the same time,it shows the internal harmony between analysis and geometry.