Question

(i) sin A. sin (B-C) + sin B sin (C-A) sin C. sin (A-B) = 0
3. Prove the following:
(i) Cos A sin (B-C)+sin B sin (C-A) + cos C sin (A-B) = 0​

Answer 1

Answer:

I) Use sin formula

sinA/a =sinB/b = sinC/c=k

sinA=ak,sinB=bk,sinc=ck

Also, sin(A−B)=sinA..cosB−cosA−sinB

=akcosB−cosA.bk

k(acosB−bcosA)

Similarity, sin(B−C)=k(bcosC−ccosB)

sin(C−A)=k(ccosA−acosc)

LHS=asin(B−C)+bsin(C−A)+csin(A−B)

=ak(bcosc−cosB)+bk(ccosA−acosC+ck(acosB−bcosA)

=k(bccosA−bccosA)

+k(accosB−accosB(abcos−abcosc)

=0+0+0

=R.H.S

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