Question

If A+B+C=π, prove that sin²A/2+sin²B/2+sin²C/2=1-2sinA/2 sinB/2 sinC/2

Answer 1

given A+B+C=180
LHS      SIN²A/2+SIN²B/2+SIN²C/2
            1/2(2SIN²A/2+2SIN²B/2+2SIN²C/2)
            1/2(1-COSA+1-COSB+1-COSC)
            1/2(3-(COSA+COSB+COSC)
            1/2(3- (2COS (A+B/2)COS(A-B/2)+COSC)
            1/2(3- (2COS(90-C/2)COS(A+B/2)+C0SC)
            1/2(3- (2SINC/2COS(A-B/2)+1-2SIN²C/2)
            1/2(3-1-2SINC/2(COS(A+B/2)-SINC/2)
            1/2(2-2SINC/2(COS(A-B/2)-SIN (90-A+B/2)  (SIN90-Ф)= COSФ)
            1/2(2-2SINC/2(COS(A-B/2)-COS(A+B/2)
            1/2(2-2SINC/2(2SINA/2SINB/2)
            1/2(2-4SINA/2SINB/2SINC/2)
            1-2SINA/2SINB/2SINC/2
            RHS       

Answer 2

given A+B+C=180

LHS      SIN²A/2+SIN²B/2+SIN²C/2

           1/2(2SIN²A/2+2SIN²B/2+2SIN²C/2)

           1/2(1-COSA+1-COSB+1-COSC)

           1/2(3-(COSA+COSB+COSC)

           1/2(3- (2COS (A+B/2)COS(A-B/2)+COSC)

           1/2(3- (2COS(90-C/2)COS(A+B/2)+C0SC)

           1/2(3- (2SINC/2COS(A-B/2)+1-2SIN²C/2)

           1/2(3-1-2SINC/2(COS(A+B/2)-SINC/2)

           1/2(2-2SINC/2(COS(A-B/2)-SIN (90-A+B/2)  (SIN90-Ф)= COSФ)

           1/2(2-2SINC/2(COS(A-B/2)-COS(A+B/2)

           1/2(2-2SINC/2(2SINA/2SINB/2)

           1/2(2-4SINA/2SINB/2SINC/2)

           1-2SINA/2SINB/2SINC/2

           RHS