2009/009) Prove that tan (A+B) = (tan A+ tan B)/(1-tan A tan B)

Proof:

We know that

the argument of a complex number z = x + yi is the angle to the real axis

then if we take the 1st number 1 + ai and argument is A and 2nd number 1+bi and argument is B

then a = tan A and b = tan B

1st number = 1 + ai = 1 + i tan A
2nd number = 1+ bi = 1 + i tan B


When we multiply the arguments add so angle is A + B

Now (1+ ai)(1+bi) = (1-ab + i(a+b))

As argument is A+B hence

tan (A+B) = (a+b)/(1-ab)= (tan A + tan B)/(1- tan A tan B) (note tan(arg) = y/x