We know that #tan(a+b) = (tan(a) + tan(b))/(1-tan(a)tan(b))# and #tan(a-b) = (tan(a) - tan(b))/(1+tan(a)tan(b))#. So we apply those formulas where they're needed.
#tan(x) + tan(120+x) - tan(120-x)= 0 iff tan(x) + (tan(120) + tan(x))/(1-tan(120)tan(x)) - (tan(120) - tan(x))/(1+tan(120)tan(x)) = 0#.
Here, you switch the minus on the denominator of the 2nd fraction so you have the same denominator everywhere and everything gets canceled!
#tan(x) + tan(120+x) - tan(120-x)= 0 iff tan(x) + (tan(120) + tan(x) - tan(120) - tan(x))/(1-tan(120)tan(x)) = 0 iff tan(x) = 0 iff x in piZZ#
