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If cos(A+B)sin(C-D)=cos(A-B)sin(C-D)then show that tanA.tanB.tanC.tanD=0

If cos(A+B)sin(C-D)=cos(A-B)sin(C-D)then show that tanA.tanB.tanC.tanD=0

Grade:11

1 Answers

Arun
25750 Points
5 years ago
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)

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