\[A, \: B, \: C\]
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.