Assume that we have a triangle with angles A, B, C. so A + B + C = πWhat is the maximum value of ( 1 − tan ⁡ A ) ( 1 − tan ⁡ B ) ( 1 − tan ⁡ C )?My try: I know that given A + B + C = π we have that tan ⁡ A + tan ⁡ B + tan ⁡ C = tan ⁡ A tan ⁡ B tan ⁡ CUsing the equalities above, I came to the result ( 1 − tan ⁡ A ) ( 1 − tan ⁡ B ) ( 1 − tan ⁡ C ) = 1 − 2 ( tan ⁡ A + tan ⁡ B + tan ⁡ C ) + ( tan ⁡ A tan ⁡ B + tan ⁡ A tan ⁡ C + tan ⁡ B tan ⁡ C )and this is where I'm stuck, using the second equality I didn't get anything helpful.I'm trying to solve this without using differentiation. any hints would be appreciated.

Question

Assume that we have a triangle with angles A, B, C. so   A  +  B  +  C  =  πWhat is the maximum value of   (  1  −  tan  ⁡  A  )  (  1  −  tan  ⁡  B  )  (  1  −  tan  ⁡  C  )?My try: I know that given   A  +  B  +  C  =  π we have that  tan  ⁡  A  +  tan  ⁡  B  +  tan  ⁡  C  =  tan  ⁡  A  tan  ⁡  B  tan  ⁡  CUsing the equalities above, I came to the result  (  1  −  tan  ⁡  A  )  (  1  −  tan  ⁡  B  )  (  1  −  tan  ⁡  C  )  =  1  −  2  (  tan  ⁡  A  +  tan  ⁡  B  +  tan  ⁡  C  )  +  (  tan  ⁡  A  tan  ⁡  B  +  tan  ⁡  A  tan  ⁡  C  +  tan  ⁡  B  tan  ⁡  C  )and this is where I&#039;m stuck, using the second equality I didn&#039;t get anything helpful.I&#039;m trying to solve this without using differentiation. any hints would be appreciated.

Garagozzodgc · Accepted Answer

The maximum value is infinity. Take  A  =      π    2    −  ϵ  ,  B  =      π    3    +  ϵ  ,  C  =      π    6  for   ϵ  &amp;gt;  0 arbitrarily small.The minimum value is also infinity. Take  A  =      π    2    +  ϵ  ,  B  =      π    3    −  ϵ  ,  C  =      π    6  for   ϵ  &amp;gt;  0 arbitrarily small.

Maximum value of (1−tan A)(1−tan B)(1−tan C)

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