Wednesday, July 3, 2019

sequences and series - If a,b,c are in arithmetic progression, prove that...



If a,b,c are in an Arithmetic Progression (AP), prove that

sincsinacosacosc=cotb.



I tried setting a,b,c as (a),(a+d),(a+2d) respectively as they are in an AP.It does not work at all. Is there any other method???


Answer



The trick is let a=Ad b=A and c=A+d.
sincsinacosacosc=sin(A+d)sin(Ad)cos(Ad)cos(A+d)
sinAcosdcosAsindsinAcosdcosAsindcosAcosdsinAsindcosAcosdsinAsind



Which simplifies to cosAsinA
cosAsinA=cotA

cotA=cotb



Quod erat demonstrandum.


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