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Question

If A,B,C be the angles of triangle ABC,such that A = π4 and tanBtanC= λ, then prove that
λ<(21)2orλ>(2+1)2

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Solution

Since A=π4B+C=3π4
tanBtanC=λtanBtan(3π4B)=λ
or tanB(1tanB)1tanB=λtan3π4=1
or tan2B+(1λ)tanB+λ=0
Δ0(1λ)24λ0
or λ26λ+10
Its roots are a=322,b=3+22
(λa)(λb)0,a<b
λaorλb
where a=(21)2,b=(2+1)2

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