Question

If a+b+c=180, prove that sin²a/2-sin²b/2+sin²c/2=1-2cosa/2sinb/2cosc/2

Answer 1

Answer:

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 Universities