Friday, 27 September 2019

sequences and series - If sides a, b, c of triangleABC are in arithmetic progression, then 3tanfracA2tanfracC2=1




If sides a, b, c of ABC (with a opposite A, etc) are in arithmetic progression, then prove that
3tanA2tanC2=1




My attempt:




a, b, c are in arithmetic progression, so
2b=a+c2sinB=sinA+sinC2sin(A+C)=2sinA+C2cosAC22sinA+C2cosA+C2=sinA+C2cosAC22cosA+C2=cosAC2


Answer




Expand your last line: 2(cosA2cosC2sinA2sinC2)=(cosA2cosC2+sinA2sinC2)
and your result is immediate after a cancellation.


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