Prove-arctan1-arctan2-arctan3-pi-

Question Number 151943 by Huy last updated on 24/Aug/21

Prove \arctan 1 + \arctan 2 + \arctan 3 = π

Commented by puissant last updated on 24/Aug/21

x = a r c t a n 1, y = a r c t a n 2, z = a r c t a n 3

p = x + y

\Rightarrow t a n (p) = t a n (x + y) = \frac{t a n x + t a n y}{1 - t a n x t a n y}

= \frac{1 + 2}{1 - 2} = - 3

t a n (z + p) = \frac{t a n z + t a n p}{1 - t a n z t a n p} = \frac{3 - 3}{1 + 9} = 0

t a n (p + z) = t a n (x + y + z) = 0 = t a n π

\Rightarrow a r c t a n 1 + a r c t a n 2 + a r c t a n 3 = π .

Answered by Olaf_Thorendsen last updated on 24/Aug/21

\arctan a = \frac{π}{2} - \arctan \frac{1}{a}

\arctan a - \arctan b = \arctan (\frac{a - b}{1 + a b})

x = \arctan 1 + \arctan 2 + \arctan 3

x = \arctan 1 + \frac{π}{2} - \arctan \frac{1}{2} + \frac{π}{2} - \arctan \frac{1}{3}

x = π + \arctan (\frac{1 - \frac{1}{2}}{1 + 1 . \frac{1}{2}}) - \arctan \frac{1}{3}

x = π + \arctan \frac{1}{3} - \arctan \frac{1}{3}

x = π