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When is the Identity tanAtanBtanC=tanA+tanB+tanC Valid?

If  
\[A, \: B, \: C\]
  are the angles of a triangle, then  
\[A+B+C=180\]
  and  
\[tanAtanBtanC=tanA+tanB+tanC\]
.
Is it true though that if  
\[tanAtanBtanC=tanA+tanB+tanC\]
. then  
\[A, \: B, \: C\]
  are then angles of a triangle?
Not it is not. Then tangent function repeats every 180 degrees, so we can add multiples of 180 to any of  
\[A, \: B, \: C\]
  or all of then and the identity would still hold. The necessary and sufficient condition for  
\[A, \: B, \: C\]
  to satisfies the identity is than  
\[A+B+C=180k,\]
  where  
\[k\]
  is any whole number, positive, negative or zero.

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