Prove that a triangle $$ABC$$ is equilateral if and only if $$\tan A + \tan B + \tan C = 3\sqrt 3 $$.

If $${\omega}$$ is an imaginary cube root of unity, then $${(1\, + \omega \, - {\omega ^2})^7}$$ equals

Let $${A_0}{A_1}{A_2}{A_3}{A_4}{A_5}$$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $${A_0}{A_1},{A_0}{A_2}$$ and $${A_0}{A_4}$$ is

If in a triangle $$PQR$$, $$\sin P,\sin Q,\sin R$$ are in $$A.P.,$$ then