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
• B. The sum of any two of A, B, C is equal to third
• C. A + B + C must be an integral multiple of π
• D. None of these

Answer 1

The correct option is C

A + B + C must be an integral multiple of π

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

We have seen when A + B + C = $\pi$

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

A + B = $\pi$ - C

tan (A + B) = tan ($\pi$ - C)

$\frac{tanA+tanB}{1-tanAtanB}$= - 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 = n$\pi$ where n $\in$ I

So, A + B + C = n$\pi$

A + B = n$\pi$ - C

tan (A + B) = tan (n$\pi$ - C)

$\frac{tanA+tanB}{1-tanAtanB}$= - 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.