Question

In `DeltaABC` if sinA.sin(B-C)= sinC.sin(A-B),then-`(A!=B!=C)` (A) tanA, tanB, tanC are in arithmetic progression (B) cotA, cotB, cotC are in arithmetic progression (C) cos2A, cos2B, cos2C are in arithmetic progression (D) sin2A. sin2B. sin2C are in arithmetic progression

Answer 1

A+B+C=180
A=180-(B+C)
C=180-(A+B)
sin(180-(B+C))sin(B_C)=sin(180-(A+B)sin(A-B)
sin(B+C)sin(B-C)=sin(A+B)sin(A-B)
`2sin^B-2sin^2C=2sin^2A-2sin^2B`
`4sin^2B-2=2sin^2A+2sin^2C-2`
`2sin^(-2)B-1+2sin^(-2)B-1=2sin^(-2)A-1+2sin^(-2)C-1 `
cos2B+cos2B=cos2A=Cos2C
2cos2B=Cos2A+cos2C.