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Question

In ΔABC
a2,b2,c2 are in A.P.
Show cotA,cotB,cotC are in A.P

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Solution

Given a2,b2,c2 or in A.P
b2a2=c2b2
K2sin2BK2sin2A=K2sin2CK2sin2B
sin(B+A).sin(BA)=sin(C+B).sin(CB)
sin(πC).sin(BA)
=sin(πA)sin(CB)
sinC.sin(BA)=sinA.sin(CB)
sin(BA)sinA×sinB=sin(CB)sinC×sinB
sinBcosAsinAsinBcosBsinAsinA.sinB=sinCcosBsinC.sinBcosC.sinBsinC.sinB
cotAcotB=cotBcotC
cotA,cotB,cotC or in AP
sinAa=sinBb=sinCc=K
A+B+C=π

1448284_843873_ans_c73bcdf477044a288a2f20743d0cba44.png

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