原式=sinAsinB/sinC+sinBsin(A+B)/sinA+sinAsin(A+B)/sinB

=sinAsinB/sinC+sinBsinBsin(A+B)/sinAsinB+sinAsinAsin(A+B)/sinBsinA (通分）

=sinAsinB/sinC+sin(A+B)(sin²B+sin²A）/sinAsinB

=sinAsinB/sinC+sin(A+B)/sinAsinB

由于sinA，sinB大于0，可运用均值不等式，即

sinAsinB/sinC+sin(A+B)/sinAsinB≥2根号{（sinAsinB/sinC）*sin(A+B)/sinAsinB}

=2

所以原式的最小值为2

可能写的不详细，你自己可以看看，当参考