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Crystal Wheeler 2022-07-02 Answered
Proof that a tan x + a cot x 2 a where 1 2 a 1 and 0 x π 4
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Answers (1)

eurgylchnj
Answered 2022-07-03 Author has 14 answers
If we set tan ( x ) = z we have z [ 0 , 1 ] and we may find the maximum of
f ( a ) = a z 1 + a 1 z 1
over the interval a [ 1 2 , 1 ] . Since the given function is convex (it is the sum of two convex functions) the maximum is attained at the boundary. At a=1 we have f(a)=2 and at a = 1 2 we have
f ( a ) = 2 2 z + 2 2 1 / z 2
hence f ( a ) 2 as wanted.

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