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Question

If tan A + tan B + tan C = tan A .tan B . tan C then


A

A, B, C must be angles of a triangle

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B

The sum of any two of A, B, C is equal to third

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C

A + B + C must be an integral multiple of

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D

None of these

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Solution

The correct option is C

A + B + C must be an integral multiple of


It is given that tan A + tan B + tan C = tan A .tan B . tan C

We have seen when A + B + C = π

We get the relation tan A + tan B + tan C = tan A .tan B . tan C

A + B = π - C

tan (A + B) = tan (π - C)

tanA+tanB1tanAtanB= - tan C

tan A + tan B = - tan C + tan A . tan B . tan C
tanA+tanB+tanC=tanA.tanB.tanC

A + B + C = π or A, B, C must be angles of triangle.
Is that the only condition when this identity is true?
We observe that when A + B + C = nπ where n ϵ I

So, A + B + C = nπ

A + B = nπ - C

tan (A + B) = tan (nπ - C)

tanA+tanB1tanAtanB= - tan C
{for any integral values of n}

tan A + tan B + tan C = tan A . tan B . tan C

So, option A is NOT correct option C is the correct option.

Option A is one special case of option C.


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