Question

if cos(A-B)/cos(A+B) + cos(C-D)/cos(C+D) =0 prove that tanAtanBtanCtanD =-1

Answer 1

cos(A-B)/cos(A+B)=-cos(C-D)/cos(C+D)
or, cos(A-B)+cos(A+B)/cos(A-B)-cos(A+B)=-cos(C-D)+cos(C+D)/-cos(C-D)-cos(C+D)
{by componendo and dividendo}
or, (cosAcosB+sinAsinB)+(cosAcosB-sinAsinB)/(cosAcosB+sinAsinB)-(cosAcosB-sinAsinB)=-(cosCcosD+sinCsinD)+(cosCcosD-sinCsinD)/-(cosCcosD+sinCsinD)-(cosCcosD-sinCsinD)
or,2cosAcosB/2sinAsinB=-2sinCsinD/-2cosCcosD
or,cotAcotB=tanCtanD
or,1/tanA×1/tanB=tanCtanD
or,1=tanAtanBtanCtanD
or,tanAtanBtanCtanD=1