Question

Can you prove that?

`sin(S-A)*sin(S-B)+ sin(S-C)*sin(S)= sinA*sinB if A+B+C=2S`

Answer 1

Please see below

Explanation:

Things to know:-`2sinAsinB=cos(A-B)-cos(A+B) and cos(-A)=cosA`

`LHS=sin(S-A)sin(S-B)+sin(S-C)sinS`

`=1/2[2sin(S-A)sin(S-B)+2sin(S-C)sinS]`

`=1/2[cos(cancel(S)-Acancel(-S)+B)-cos(S-A+S-B)+cos(cancel(S)-Ccancel(-S))-cos(S-C+S)]`

`=1/2[cos(B-A)-cos(2S-A-B)+cos(-C)-cos(2S-C)]`

`=1/2[cos(B-A)-cos(cancel(A)cancel(+B)+Ccancel(-A)cancel(-B))+cosC-cos(A+Bcancel(+C)cancel(-C))]`

`=1/2[cos(A-B)-cos(A+B)cancel(+cosC)cancel(-cosC)]`

`=1/2[2sinAsinB]=sinAsinB=RHS`