Sea: a = 30 º, b = 80 º y c = 70 º,
tan(a)*tan(b) + tan(b)*tan(c) + tan(a)*tan(c) ≠ tan(a)*tan(b)*tan(c)
tan(30)*tan(80) + tan(80)*tan(70) + tan(30)*tan(70) ≠ tan(30)*tan(80)*tan(70)
20,4422916517 ≠ 8,9961095083
Es decir, la ecuación es incorrecta.
Por otro lado,
a + b = 180 - c
Entonces:
a + b = 180 - c (aplicando tangente a ambos lados de la ecuación)
tan(a + b) = tan(180 - c)
Usando la siguiente identidad trigonométrica:
tan(A + B) = [tan(A) + tan(B)] / [1 - tan(A)*tan(B)]
Se tiene:
[tan(a) + tan(b)] / [1 - tan(a)*tan(b)] = [tan(180) + tan(-c)] / [1 - tan(180)*tan(-c)]
ademas: tan(180) = 0 y tan(-c) = -tan(c),
tan(a) + tan(b) = -tan(c)[1 - tan(a)*tan(b)]
tan(a) + tan(b) = -tan(c) +tan(a)*tan(b)*tan(c)
En otras palabras:
tan(a) + tan(b) + tan(c) = tan(a)*tan(b)*tan(c)
Evaluando los valores arbitrarios que me dí en un principio:
tan(30) + tan(80) + tan(70) = tan(30)*tan(80)*tan(70)
8,9961095083 = 8,9961095083
