9a²+9b²-19c²=0→(a²+b²)/ c²=19/9
由正弦公式有(sinA*sinB)/sin2C=（a/c）*(b/c)
A+B+C=Л→A+B=Л-C→
sinAcosB+sinBcosA= sin(A+B) =sin(Л-C)= sinC
由余弦定理有 cosC=(a²+b²-c²)/(2*a*b)
根据以上条件：
cotC/(cotA+cotB)
=(cosC/sinC)/((cosA/sinA)+(cosB/sinB))
=(cosC/sinC)/(( sinAcosB+sinBcosA)/(sinA*sinB))
=( cosC/sinC)/(sinC/( sinA*sinB))
=cosC*((sinA*sinB)/sin2C)
=cosC*（a/c）*(b/c)
=[(a²+b²-c²)/(2*a*b)]*[a*b/c²]
=(a²+b²-c²)/ (2* c²)
=(1/2)*(( a²+b²)/ c² -1)
=(1/2)*(19/9-1)
=5/9
(这样看不太直观,写成分式就好了,还有sin2C是指sinC * sinC,*是指乘法)