Question

If cos (A+B)/cos (A-B) = sin (C+D)/sin (C-D) prove that tanA tanB tan C + tanD=0

Answer 1

cos(A+B)/cos(A-B)=sin(C+D)/sin(C-D)
using devidendo-componendo process,
[cos(A-B)-cos(A+B)]/[cos(A-B)+cos(A+B)]=[sin(C-D)-sin(C+D)]/[sin(C-D)+sin(C+D)]
or, 2sinAsinB/[cos(A+B)+cos(A-B)]=-[sin(C+D)-sin(C-D)]/[sin(C+D)+sin(C-D)]
or, 2sinAsinB/2cosAcosB=-2cosCsinD/2sinCcosD
or, tanAtanB=-cotCtanD
or, tanAtanB/cotC=-tanD
or, tanAtanBtanC=-tanD
or, tanAtanBtanC+tanD=0 (proved)