Analysis: Construct a quadratic function and prove it by discriminant.

Proof: Construct a quadratic function about X.

f(x)= x2-2x(zcosB+ycosC)+y2+z2-2yzcosA

The quadratic coefficient of f (x) is greater than zero,

To make f(x)≥0 hold for all x∈R constants, we only need to prove the discriminant δ ≤ 0 of f(x).

∑δ= 4(zcosB+ycosC)2-4(y2+z2-2 yzcosa)

=-4z 2s in 2 b-4y 2s in 2c+8 zycosbcosc+8 yzcosa

=-4z 2s in 2 b-4y 2s in 2c+8zysinBsinC

=-4(zsinB-ysinC)2≤0

Therefore, the original inequality holds.