Can you prove that?

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

1 Answer
Jul 15, 2018

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