Question

How do you prove that ArcTan(1) + ArcTan(2) + ArcTan(3) = π?

Answer 1

Prove that (arctan (1) + arctan (2) + arctan (3) = pi)

Explanation:

Call artan (1) = x; arctan (2) = y; and arctan (3) = z
Apply the trig identity: `tan (a + b) = (tan a + tan b)/(1 - tan a.tan b)`
First evaluate tan u = tan (x + y);
`tan u = tan (x + y) = (tan x + tan y)/(1 - tan x.tan y) = (1 + 2)/(1 - 2) = - 3`
Next, evaluate tan (z + u)
`tan (z + u) = (tan z + tan u)/(1 - tan z.tan u) = (-3 + 3)/(1 - 9) = 0`
Finally: tan (u + z) = tan (x + y + z) = 0 = `tan pi`, therefor:
arctan (1) + arctan (2) + arctan (3) = x + y + z = `pi`