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Sin²A/2+sin²B/2+sin²C/2=1-2sinA/2sinB/2sinC/2 Need to prove this formula

Sin²A/2+sin²B/2+sin²C/2=1-2sinA/2sinB/2sinC/2
Need to prove this formula
 

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Grade:12

1 Answers

Arun
25750 Points
5 years ago
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       
 

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