tan tan tan

That was a question I saw at Polymathematics raised:

Prove that if A, B, and C are the 3 angles of a triangle,

tan A + tan B + tan C = (tan A)(tan B)(tan C).

There is demonstration for this fact using complex numbers say A + B + C = 180

(cosA + i sinA)*(cosB + i sinB)*(cosC + i sinC)=-1

So this product is real number dividing each factor by the cos we have

(1 + i tanA)*(1 + i tanB)*(1 + i tanC) is real

By foiling and since the imaginary part is 0 we get that identity.

That was the nicest way I could show that identity.

My interest in it was that it can be used to show Hieron's formula, as I was trying to find nicer ways to show it I got the way above.