令cosx=k,cosy=l,cosz=m
则S=tan{x/2}+tan{y/2}+tan{z/2}
={(1-k)/(1+k)}^{1/2}+{(1-m)/(1+m)}^{1/2}+{(1-l)/(1+l)}^{1/2}
={(m+l)/(m+n+l+n)}^{1/2}+{(m+n)/(m+n+l+l)}^{1/2}+{(n+l)/(m+n+l+m)}^{1/2}
令a=k+l,b=l+m,c=k+m
则abc为三角形三边
且S=sum{a/(b+c)}^{1/2}
易证{a/(b+c)}^{1/2}>=(2a)/(a+b+c)
所以S>=2
且x=y=pi/2,z=0时,S=2,
所以最小值为2