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If A+B+C=180Prove that Sin^2A/2 +sin^2B/2+sin^2C/2=1_2sinA/2sinB/2sinC/2

If A+B+C=180Prove that Sin^2A/2 +sin^2B/2+sin^2C/2=1_2sinA/2sinB/2sinC/2

Grade:11

3 Answers

Vikas TU
14149 Points
4 years ago
Dear Student,
Taking LHS:
=Sin^2 A/2+sin^2 B/2+sin^2 C/2
=1-cos^2 A/2 +sin^2 B/2+sin^2 C/2
=1-(cos(A+B)/2 cos(A-B)/2) + sin^2 C/2
= 1-(cos(pi-C)/2 cos(A-B)/2)+sin^2C/2
= 1+sin C/2(sin C/2 -cos(A-B)/2)
=1+sin C/2(sin(pi –(A+B)/2 -cos(A-B)/2)
=1+sin C/2(cos (A+B)/2-cos(A-B)/2)
=1+sin C/2(-2sinA/2 sinB/2) = 1-2sinA/2 sin B/2 sin C/2 =RHS [hence proved].
Cheers!!
Regards,
Vikas (B. Tech. 4th year
Thapar University)
Saurabh Singh
13 Points
3 years ago
Given that :-Sin²a/2 + sin²b/2 + sin²c/2(1 - Cosa + 1- cosb +1 - cos c)/2(3 - ( Cosa +cosb + cosc))/2(3 -(1 +4 sin²a/2 sin²b/2sin²c/2))/2(2 - 4 sin²a/2 sin²b/2sin²c/2)/21-2sin²a/2 sin²b/2sin²c/2
Kushagra Madhukar
askIITians Faculty 629 Points
one year ago
Dear student,
Please find the answer to your question.
 
Taking LHS:
=Sin^2 A/2+sin^2 B/2+sin^2 C/2
=1-cos^2 A/2 +sin^2 B/2+sin^2 C/2
=1-(cos(A+B)/2 cos(A-B)/2) + sin^2 C/2
= 1-(cos(pi-C)/2 cos(A-B)/2)+sin^2C/2
= 1+sin C/2(sin C/2 -cos(A-B)/2)
=1+sin C/2(sin(pi –(A+B)/2 -cos(A-B)/2)
=1+sin C/2(cos (A+B)/2-cos(A-B)/2)
=1+sin C/2(-2sinA/2 sinB/2) = 1-2sinA/2 sin B/2 sin C/2 =RHS [hence proved].
 
Thanks and regards,
Kushagra

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