Question

Prove that:
`(sin(A-C)+2sinA+sin(A+C))/(sin(B-C)+2sinB+sin(B+C))=(sinA)/(sinB)`

Answer 1

LHS =`(sin(A-C)+2sinA+sin(A+C))/(sin(B-C)+2sinB+sin(B+C))`
`=(sin(A+C)+sin(A-C)+2sinA)/(sin(B+C)+sin(B-C)+2sinB)`
`(2sin(A+C+A-C)/(2).cos(A+C-A+C)/(2)+2sinA)/(2sin((B+C+B-C)/2).cos(B+C-B+C)/(2)+2sinB)`
`=(2sinAcosC+2sinA)/(2sinBcosC+2sinB)`
`(sinA(2cosC+2))/(sinB(2cosC+2))`
`=(sinA(2cosC+2))/(sinB(2cosC+2))`
`=(sinA)/(sinB)` =RHS Hence Proved.