sin (b+2c)+sin(c+2a)++ sin(a+2b) =4sin(b-c/2) sin(c-a/2) sin(a-b/2) when a+b+c =pie - Maths - Trigonometric Functions

Question

sin (b+2c)+sin(c+2a)++ sin(a+2b) =4sin(b-c/2) sin(c-a/2)
sin(a-b/2) when a+b+c =pie

Anuradha Sharma · Accepted Answer

Given ,A+B+C=180, 
1and LHS is,sinA+2B+sinB+2C+sinC+2A=sin180-C-B+sin180-A-C+sin180-B-A from equation 1=sinC-B+sinA-C+sinB-A=2sinC-B2cosC-B2+2sinB-C2cos2A-B-C2As we know that sinx+siny = 2sinx+y2
cosx-y2=2sinB-C2-cosC-B2+cos2A-B-C2=2sinB-C22sinA-B2sinC-A2As we know that cosx-cosy = 2sinx+y2
siny-x2=42sinB-C2sinA-B2sinC-A2

sin (b+2c)+sin(c+2a)++ sin(a+2b) =4sin(b-c/2). sin(c-a/2).sin(a-b/2) when a+b+c =pie