If a,b,c are in HP . then, ab+bc=2ac.

using this prove that

sin²(A/2).sin²(B/2)+sin²(B/2).sin²(C/2)= 2sin²(A/2).sin²(C/2)

Let assume later part is correct hence we have to prove a, b, c are in hp

X/2= (1-cosX)/2Use the cosine law to indulge the sides a,b,c into the eqn.Simplify to get the answer.²C/2...(AP)Write the average of this AP.Convert Sin²B/2, 1/Sin²A/2, 1/Sin²Write the AP average term in context to the HP given :1/Sin