I have solved this problem in a way, rather "inspired". I would like to have a solution found an easier way but I was unable so far.
Let $A,B,C$ the angles of a triangle $\triangle {ABC}$; prove that $$(\sin^2 A+\sin^2 B+\sin^2 C)\le \frac14 \left(\frac{1}{\sin A}+\frac{1}{\sin B}+\frac{1}{\sin C}\right) (\sin A+\sin B+\sin C)$$