Question

if cos(A+B)sin(C+D)=cos(A-B).sin(C-D) prove that cotA.cotB.cotC=cotD

Answer 1

Answer:

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)

Answer 2

your solution is in attachment.

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