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Find μ if tanA + tanB + tanC= μtanAtanBtanC. Find the maximum value of cotAcotBcotC.

Find μ if tanA + tanB + tanC= μtanAtanBtanC. Find the maximum value of cotAcotBcotC.

Grade:11

1 Answers

Aditya Gupta
2081 Points
5 years ago
assuming A, B C are the angles of  a triangle, μ=1
now, for cotAcotBcotC=1/tanAtanBtanC to be maximum, tanAtanBtanC has to be minimum positive value. let tanAtanBtanC=p
by AM-GM ineq, we have
(tanA + tanB + tanC)/3>=(tanAtanBtanC)^1/3
or p/3>=p^1/3
or p^2/3>=3
so p>=3√3. 
so p min = 3√3 which occurs when A=B=C=60Degre
 maximum value of cotAcotBcotC is 1/3√3

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